#8222665

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
prime radical	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(3,7)(5,8)(25,26)(27,34)(28,33)(29,30)(31,35)(32,38)(36,37)(39,40);(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,27)(26,32)(28,36)(29,37)(30,39)(31,38)(33,40)(34,35);(1,3)(2,5)(4,7)(6,8)(10,14)(15,21)(16,19)(22,23)(25,28,31,30)(26,33,35,29)(27,36,38,39)(32,40,34,37);(1,2)(3,5)(4,6)(7,8)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,29)(26,28)(27,37)(30,35)(31,33)(32,36)(34,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,12,13,20)(10,16,14,19)(11,17,18,24)(15,22,21,23)(25,30,31,28)(26,29,35,33)(27,36,38,39)(32,40,34,37);(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,27)(26,34)(28,39)(29,37)(30,36)(31,38)(32,35)(33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8129861);