#8461212

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

4973 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8368408);