#8599124

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

4779 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)(4,5)(6,7)(17,18)(19,25)(20,30)(21,22)(23,26)(24,27)(28,32)(29,31);(1,3,6,5)(2,4,7,8)(10,13)(14,16)(17,19,24,29)(18,25,27,31)(20,26,21,28)(22,32,30,23);(2,7)(4,8)(9,10,12,13)(11,14,15,16)(17,20,24,21)(18,22,27,30)(19,28,29,26)(23,31,32,25);(2,7)(3,5)(11,15)(14,16)(17,21,24,20)(18,22,27,30)(19,26,29,28)(23,31,32,25);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,22)(18,20)(19,23)(21,27)(24,30)(25,28)(26,31)(29,32);(1,5,6,3)(2,4,7,8)(9,11)(10,14)(12,15)(13,16)(17,23)(18,26)(19,22)(20,31)(21,25)(24,32)(27,28)(29,30)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=gh,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, 8506320);