#72318308

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2952015
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#419812	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=fh,c^2=hi,b^a=bf,c^b=cfg,d^a=dfhij,d^b=dfhij,d^c=dij,e^a=egi,e^d=eij,f^b=fj,f^c=fj,g^a=gj,g^d=gj,h^a=hj,h^b=hj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318468, #72319027, #72319065, …
maximal normal subgroup of	#72318468, #72319027, #72319065, …

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