#465201

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1107 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:

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

Polycyclic presentation:

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

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, 372397);