#452535

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

919 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)(9,10,14,19)(11,20)(12,25,27,34)(13,15,29,18)(16,24)(17,28,33,26)(21,36)(22,32)(23,30)(31,37)(35,39)(38,40);(1,3)(2,5)(4,7)(6,8)(10,15,19,18)(12,26,27,28)(16,30,20,32)(17,33)(21,37,38,39)(25,34)(31,35)(36,40);(3,7)(5,8)(9,11,13,22,14,24,29,23)(10,16,15,30,19,20,18,32)(12,21,28,39,27,38,26,37)(17,31,25,36,33,35,34,40);(9,12,13,28,14,27,29,26)(10,17,15,25,19,33,18,34)(11,21,22,39,24,38,23,37)(16,31,30,36,20,35,32,40)

Polycyclic presentation:

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

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, 359731);