#416364

order	512
nilpotency class	3
isoclinism	isoclinic class of #394192
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6,9,10)(7,11,8,12)(13,14,19,22,18,23,17,30)(15,24,31,34,26,38,27,20)(16,21,28,41,29,36,32,35)(25,37,40,33,39,44,42,43);(1,2)(3,4)(6,10)(11,12)(13,15,17,27,18,26,19,31)(14,20,22,24,23,34,30,38)(16,25,28,40,29,39,32,42)(21,33,35,37,36,43,41,44);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,28,18,29,19,32)(14,21,22,35,23,36,30,41)(15,25,27,40,26,39,31,42)(20,33,24,37,34,43,38,44);(5,8,9,7)(6,11,10,12)(15,26)(20,34)(24,38)(25,39)(27,31)(33,43)(37,44)(40,42)

Polycyclic presentation:

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

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, 323560);