#452628

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

631 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)(9,10,16,21,15,20,14,30)(11,22,25,33,26,17,31,36)(12,27,29,19,28,35,13,18)(23,37,24,34,38,40,39,32);(1,3)(2,5)(4,7)(6,8)(9,11,14,25,15,26,16,31)(10,17,30,36,20,22,21,33)(12,23,13,24,28,38,29,39)(18,32,35,37,19,34,27,40);(3,7)(5,8)(9,12,14,13,15,28,16,29)(10,18,21,27,20,19,30,35)(11,23,25,24,26,38,31,39)(17,32,33,40,22,34,36,37);(9,13,14,28,15,29,16,12)(10,19,30,27,20,18,21,35)(11,24,25,38,26,39,31,23)(17,32,36,37,22,34,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359824);