#6443241

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4993 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)(9,10,15,20)(11,21,25,31)(12,26,27,34)(13,19,14,18)(16,24,32,23)(17,28,33,29)(22,30,36,40)(35,37,39,38);(1,2,3,4)(5,6)(7,8)(9,11,14,24,15,25,13,23)(10,16,19,31,20,32,18,21)(12,22,29,38,27,36,28,37)(17,30,26,39,33,40,34,35);(6,8)(9,12,14,29,15,27,13,28)(10,17,19,26,20,33,18,34)(11,22,24,38,25,36,23,37)(16,30,31,39,32,40,21,35);(6,8);(12,27)(17,33)(22,36)(26,34)(28,29)(30,40)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350437);