#8900128

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

4860 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:

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

Polycyclic presentation:

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

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, 8807324);