#457177

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1384 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:

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

Polycyclic presentation:

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

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, 364373);