#72310853

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#35346	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(23,27)(35,41)(36,38)(47,48);(4,11)(6,13)(17,18)(19,27)(20,31)(21,25)(22,26)(23,30)(24,33)(28,35)(29,38)(32,39)(34,40)(36,45)(37,46)(41,43)(42,47)(44,48);(1,3)(2,6)(4,12,11,9)(5,10)(7,14,15,16)(8,13)(17,19,22,30)(18,23)(20,28,33,43)(21,29,34,45)(24,35)(25,38)(26,27)(31,41)(32,42,46,44)(36,40)(37,48)(39,47);(1,4,5,11)(2,7,8,15)(3,9,10,12)(6,14,13,16)(17,20,22,33)(18,24,26,31)(19,28,30,43)(21,32,34,46)(23,35,27,41)(25,39,40,37)(29,44,45,42)(36,47,38,48);(4,11)(7,15)(17,21)(18,25)(19,29)(20,32)(22,34)(23,36)(24,37)(26,40)(27,38)(28,42)(30,45)(31,39)(33,46)(35,47)(41,48)(43,44)

Polycyclic presentation:

2^10|c^2=gij,d^2=i,b^a=bf,c^a=cg,c^b=cfh,d^b=dhi,d^c=dh,e^c=eg,e^d=efgj,f^c=fi,f^d=fj,g^b=gij,g^d=gj,h^a=hj,h^c=hj,h^e=hij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310874, #72337579, #72340327, …
maximal normal subgroup of	#72310874, #72337579, #72340327, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.7*F.9*F.10);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);