#8701742

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

5545 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8608938);