#415231

order	512
nilpotency class	3
isoclinism	isoclinic class of #394464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,c^2=e,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,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, 322427);