#6443227

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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×#3050	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4984 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)(11,21,23,32)(12,26,29,33)(13,19,14,18)(16,25,31,24)(17,27,34,28)(22,30,36,40)(35,37,39,38);(1,2,3,4)(5,6)(7,8)(9,11,13,24,15,23,14,25)(10,16,18,21,20,31,19,32)(12,22,27,37,29,36,28,38)(17,30,33,35,34,40,26,39);(6,8)(9,12,13,27,15,29,14,28)(10,17,18,33,20,34,19,26)(11,22,24,37,23,36,25,38)(16,30,21,35,31,40,32,39);(6,8);(11,23)(16,31)(21,32)(22,36)(24,25)(30,40)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,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^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350423);