#6395459

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	#60915	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1673 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)(9,10,14,20,15,21,30,19)(11,16,17,12,23,25,33,26)(13,18,29,37,28,35,38,36)(22,31,32,24,27,34,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,17,15,23,30,33)(10,16,20,12,21,25,19,26)(13,27,29,40,28,22,38,32)(18,34,37,39,35,31,36,24);(9,12,14,25,15,26,30,16)(10,17,20,23,21,33,19,11)(13,24,29,34,28,39,38,31)(18,32,37,27,35,40,36,22);(2,5)(6,8)(9,13)(10,18)(11,22)(12,24)(14,29)(15,28)(16,31)(17,32)(19,36)(20,37)(21,35)(23,27)(25,34)(26,39)(30,38)(33,40);(1,3,4,7)(2,6,5,8)(13,28)(18,35)(22,27)(24,39)(29,38)(31,34)(32,40)(36,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,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, 6302655);