#8323182

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

5436 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=g,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, 8230378);