#8761690

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261925, #72261997, #72262168
maximal normal subgroup of	#72261925, #72261997, #72262168

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8668886);