#457237

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1334 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)(5,6)(7,8)(9,10)(11,13)(12,18)(14,22)(15,20)(16,21)(17,19)(23,31)(24,33)(25,32)(26,40)(27,39)(28,37)(29,35)(30,36)(34,38);(1,3,4,2)(9,11,14,19,16,24,15,25)(10,17,20,13,21,32,22,33)(12,26,28,38,30,23,29,39)(18,34,35,40,36,27,37,31);(2,3)(9,12,16,30)(10,18,21,36)(11,23,24,26)(13,31,33,40)(14,29,15,28)(17,34,32,27)(19,38,25,39)(20,37,22,35);(6,8)(9,13,15,17,16,33,14,32)(10,19,22,11,21,25,20,24)(12,27,29,40,30,34,28,31)(18,26,37,39,36,23,35,38)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^c=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, 364433);