#462609

order	512
nilpotency class	3
isoclinism	isoclinic class of #458125
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=e,d^2=f,e^2=i,f^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,f^c=fi,g^c=gi,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, 369805);