#8508899

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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

4891 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)(3,7)(11,12)(14,15)(17,18,22,23)(19,27,30,24)(20,21,26,25)(28,29,32,31);(1,2,4,5)(3,6,7,8)(9,10,13,16)(11,14,12,15)(18,23)(20,26)(24,27)(28,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19,22,30)(18,24,23,27)(20,28,26,32)(21,31,25,29);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,20,22,26)(18,25,23,21)(19,28,30,32)(24,29,27,31);(17,21,22,25)(18,26,23,20)(19,29,30,31)(24,28,27,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=gi,e^2=ghi,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, 8416095);