#8583319

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

5028 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)(4,6)(5,7)(11,15)(13,16)(17,18)(19,20)(21,29)(22,26)(23,24)(25,32)(27,30)(28,31);(2,4)(7,8)(9,10,12,14)(11,13,15,16)(17,19,23,27)(18,20,24,30)(21,31,32,22)(25,26,29,28);(1,3,6,5)(2,7,4,8)(9,11)(10,13)(12,15)(14,16)(18,24)(19,27)(21,32)(26,28);(1,2,6,4)(3,7,5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,28)(22,25)(23,30)(24,27)(26,32)(29,31);(1,4,6,2)(3,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,21)(18,25)(19,22)(20,28)(23,32)(24,29)(26,30)(27,31);(1,5,6,3)(2,7,4,8)(17,22)(18,26)(19,21)(20,29)(23,31)(24,28)(25,30)(27,32)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^2=g,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, 8490515);