#8182047

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

4871 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)(3,7)(4,5)(6,8)(17,18,23,26)(19,20,21,24)(22,25,30,32)(27,28,29,31);(1,3,6,5)(2,7,8,4)(9,10,13,14)(11,15,16,12)(17,19,23,21)(18,24,26,20)(22,27,30,29)(25,31,32,28);(2,8)(3,5)(10,14)(12,15)(22,30)(25,32)(27,29)(28,31);(9,11)(10,12)(13,16)(14,15)(17,20,23,24)(18,19,26,21)(22,28,30,31)(25,27,32,29);(1,4)(2,3)(5,8)(6,7)(17,21,23,19)(18,24,26,20)(22,29,30,27)(25,31,32,28);(1,5,6,3)(2,4,8,7)(9,12)(10,11)(13,15)(14,16)(17,22,23,30)(18,25,26,32)(19,27,21,29)(20,28,24,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=g,e^2=g,f^2=gh,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, 8089243);