#464332

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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1275 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)(6,9)(10,12)(13,14)(15,24)(16,29)(17,22)(18,34)(19,23)(20,26)(21,32)(25,42)(27,36)(28,41)(30,35)(31,39)(33,38)(37,43)(40,44);(5,6,8,9)(7,10,11,12)(13,15,18,27,17,26,19,28)(14,20,34,41,22,24,23,36)(16,30,32,25,31,44,33,43)(21,37,39,35,38,42,29,40);(7,11)(10,12)(13,16)(14,21)(15,25)(17,31)(18,33)(19,32)(20,35)(22,38)(23,39)(24,40)(26,43)(27,30)(28,44)(29,34)(36,42)(37,41);(2,4)(5,7,8,11)(6,10,9,12)(16,31)(21,38)(25,43)(29,39)(30,44)(32,33)(35,40)(37,42)

Polycyclic presentation:

2^9|b^2=eg,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,f^c=fi,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, 371528);