#6380438

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
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	C₂³×C₄⋊C₄
prime radical	C₂×#121	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1730 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)(6,8)(9,10,14,18,15,19,13,28)(11,16,23,31,21,30,22,37)(12,24,27,32,25,38,26,17)(20,33,35,39,36,40,34,29);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,29)(18,31)(19,30)(24,33)(25,36)(26,34)(27,35)(28,37)(32,39)(38,40);(9,12,13,26,15,25,14,27)(10,17,18,24,19,32,28,38)(11,20,22,34,21,36,23,35)(16,29,31,33,30,39,37,40);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(11,21)(12,25)(16,30)(17,32)(22,23)(24,38)(26,27)(31,37)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287634);