#6661039

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6568235);