#464251

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3154	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1754 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)(9,10,15,21)(11,23,27,36)(12,29,32,18)(13,19,31,38)(14,22,16,20)(17,28,35,26)(24,39,25,37)(30,34,40,33);(5,6,7,8)(9,11,14,26,15,27,16,28)(10,17,20,23,21,35,22,36)(12,30,31,24,32,40,13,25)(18,37,38,33,29,39,19,34);(2,4)(6,8)(9,12)(10,18)(11,24)(13,14)(15,32)(16,31)(17,33)(19,20)(21,29)(22,38)(23,37)(25,27)(26,30)(28,40)(34,35)(36,39);(9,13,16,32,15,31,14,12)(10,19,22,29,21,38,20,18)(11,25,28,40,27,24,26,30)(17,34,36,39,35,33,23,37)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 371447);