#457021

order	512
nilpotency class	3
isoclinism	isoclinic class of #456400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364217);