#7501570

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	#60453	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1919 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408766);