#8424599

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
prime radical	#34889	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8331795);