#7692433

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

4855 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,d^2=i,e^2=gi,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7599629);