#72265924

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#13956
commutator	C₂⁵	C₂³×C₄
Frattini, cosocle	C₂⁶	C₂⁴
prime radical	#7712875	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

5683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=gh,b^a=beg,c^a=cf,c^b=cfhj,d^a=dj,d^b=dj,d^c=dh,e^b=ej,e^c=ei,f^a=fi,f^d=fj,g^a=gj,g^b=gj,g^c=gj,h^a=hj,h^b=hj

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265925, #72265946, #72265975
maximal normal subgroup of	#72265925, #72265946, #72265975
maximal quotient of	#72266150

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, 1, F.5);;
gap> SetPower(coll, 2, F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);