#8431736

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

4988 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,f^2=gh,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, 8338932);