#8505298

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

5070 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412494);