#8525468

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

4762 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8432664);