#414765

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321961);