#457099

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1140 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)(9,10,13,16,15,20,14,19)(11,21,23,31,25,17,24,32)(12,26,29,36,27,34,30,18)(22,35,28,33,38,40,39,37);(1,2)(3,4)(10,16,20,19)(12,27)(17,32,21,31)(18,34,36,26)(22,38)(28,39)(29,30)(33,35,37,40);(2,4)(9,11,15,25)(10,17,20,21)(12,28,27,39)(13,24,14,23)(16,31,19,32)(18,35,36,40)(22,30,38,29)(26,37,34,33);(6,8)(9,12,14,30,15,27,13,29)(10,18,16,26,20,36,19,34)(11,22,24,39,25,38,23,28)(17,33,32,40,21,37,31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364295);