#8583147

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

5165 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:

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

Polycyclic presentation:

2^9|b^2=g,e^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, 8490343);