#8657974

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

4685 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8565170);