#8533680

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

5763 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)(19,27)(20,25)(21,23)(22,26)(24,28)(29,30)(31,32);(1,3,4,8)(2,5,6,7)(11,15)(14,16)(19,28)(20,29)(24,27)(25,30);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,28)(18,24,21,27)(20,22,29,32)(25,31,30,26);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,20)(18,25)(19,22)(21,30)(23,29)(24,31)(26,27)(28,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,18)(19,24,28,27)(20,30,29,25)(22,31,32,26);(17,22)(18,26)(19,29)(20,28)(21,31)(23,32)(24,25)(27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440876);