#8657160

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

5231 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)(3,8)(4,6)(5,7)(17,18)(19,23)(20,22)(21,24)(25,27)(26,31)(28,30)(29,32);(2,4)(7,8)(9,10,15,16)(11,13,14,12)(17,18)(19,25)(20,22)(21,24)(23,27)(26,32)(28,30)(29,31);(1,3,6,5)(2,7,4,8)(9,11,15,14)(10,12,16,13)(18,20)(23,25)(24,28)(31,32);(1,2,6,4)(3,7,5,8)(9,12,15,13)(10,11,16,14)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31);(1,4,6,2)(3,7,5,8)(9,13,15,12)(10,11,16,14)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31);(1,5,6,3)(2,7,4,8)(9,14,15,11)(10,12,16,13)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=hi,e^2=gi,f^2=gi,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, 8564356);