#8195833

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

5343 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)(3,5)(4,6)(7,8)(17,18)(19,22)(20,24)(21,29)(23,26)(25,31)(27,32)(28,30);(2,3)(6,7)(9,10,13,12)(11,15,16,14)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,32)(29,31);(4,8)(6,7)(10,12)(14,15)(17,19,23,20)(18,24,26,22)(21,27,31,28)(25,32,29,30);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,23,19)(18,24,26,22)(21,28,31,27)(25,32,29,30);(17,21,23,31)(18,25,26,29)(19,27,20,28)(22,30,24,32);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,15,16,14)(17,22,23,24)(18,19,26,20)(21,30,31,32)(25,27,29,28)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8103029);