#6389016

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	C₂×#3035	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#432 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1943 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:

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

Polycyclic presentation:

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

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, 6296212);