#8583328

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

4950 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^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, 8490524);