#8634856

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

4660 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,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,e^b=egi,e^c=egi,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, 8542052);