#462661

order	512
nilpotency class	3
isoclinism	isoclinic class of #458125
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

837 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=e,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,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, 369857);