#464177

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1769 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)(3,4)(6,7)(9,10)(11,20)(12,25)(13,17)(14,19)(15,23)(16,28)(18,21)(22,33)(24,31)(26,36)(27,35)(29,34)(30,39)(32,37)(38,40);(5,6,8,7)(9,11,21,37,13,23,14,24)(10,15,18,31,17,20,19,32)(12,26,29,22,27,39,28,38)(16,33,25,30,34,40,35,36);(6,7)(9,12)(10,16)(11,22)(13,27)(14,29)(15,30)(17,34)(18,35)(19,25)(20,36)(21,28)(23,38)(24,39)(26,37)(31,33)(32,40);(2,4)(5,7,8,6)(12,27)(16,34)(22,38)(25,35)(26,39)(28,29)(30,36)(33,40)

Polycyclic presentation:

2^9|b^2=gi,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, 371373);