#8924764

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

5122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831960);