#8195702

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

5208 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=g,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, 8102898);