#6395835

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
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	#60797	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1819 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)(3,6)(4,7)(5,8)(9,10,15,20,16,21,14,32)(11,17,23,12,24,26,18,27)(13,19,31,37,29,36,30,40)(22,33,38,25,28,35,34,39);(1,3,6,2)(4,7,5,8)(9,11,15,23,16,24,14,18)(10,17,20,12,21,26,32,27)(13,28,31,34,29,22,30,38)(19,35,37,39,36,33,40,25);(9,12,15,26,16,27,14,17)(10,18,20,11,21,23,32,24)(13,25,31,35,29,39,30,33)(19,34,37,22,36,38,40,28);(1,4,6,5)(2,7,3,8)(9,13)(10,19)(11,22)(12,25)(14,30)(15,31)(16,29)(17,33)(18,34)(20,37)(21,36)(23,38)(24,28)(26,35)(27,39)(32,40);(1,5,6,4)(2,7,3,8)(13,29)(19,36)(22,28)(25,39)(30,31)(33,35)(34,38)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=h,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,g^c=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6303031);