#8422225

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

4970 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8329421);