#8696719

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

5182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312186, #72317595, #72317624, …
maximal normal subgroup of	#72312186, #72317595, #72317624, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603915);