#8325515

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

4963 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8232711);