#8231467

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

4781 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8138663);