#6444372

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35634	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2763 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351568);