#464189

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1705 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)(5,6)(7,8)(9,10)(11,17)(12,25)(13,16)(14,19)(15,20)(18,28)(21,37)(22,31)(23,30)(24,33)(26,40)(27,36)(29,35)(32,38)(34,39);(10,16,20,19)(17,31,33,30)(18,25,36,35)(32,37,40,34);(2,4)(9,11,15,24)(10,17,20,33)(12,26,29,38)(13,23,14,22)(16,30,19,31)(18,34,36,37)(21,28,39,27)(25,40,35,32);(6,8)(9,12,14,28,15,29,13,27)(10,18,16,25,20,36,19,35)(11,21,23,38,24,39,22,26)(17,32,31,37,33,40,30,34)

Polycyclic presentation:

2^9|b^2=g,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,d^c=de,e^c=ei,h^c=hi

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, 371385);