#6378179

order	512
nilpotency class	3
isoclinism	isoclinic class of #60771
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60907	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1710 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cfi,g^c=gi,e^d=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, 6285375);