#8630883

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

4851 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8538079);