#6407572

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2404	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1553 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=dfi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6314768);