#457247

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
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×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1353 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=ei,e^2=i,f^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,f^c=fi,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, 364443);