#8587489

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

5072 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)(10,13)(11,15)(17,18,22,24)(19,25,28,23)(20,29,31,21)(26,32,30,27);(1,2)(3,8)(4,6)(5,7)(9,10,12,13)(11,14,15,16)(17,19,22,28)(18,23,24,25)(20,30,31,26)(21,27,29,32);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(18,24)(20,31)(23,25)(26,30);(2,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,32)(22,28)(24,25)(27,29)(30,31);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,21)(19,26)(22,31)(23,32)(24,29)(25,27)(28,30);(1,2)(3,5)(4,6)(7,8)(17,21)(18,20)(19,27)(22,29)(23,30)(24,31)(25,26)(28,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,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, 8494685);