#6443270

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4948 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)(9,10,16,22)(11,23,27,34)(12,28,29,37)(13,19,31,38)(14,21,15,20)(17,26,35,25)(18,30,36,24)(32,40,33,39);(1,2,3,4)(5,6)(7,8)(9,11,14,25,16,27,15,26)(10,17,20,23,22,35,21,34)(12,13,30,32,29,31,24,33)(18,19,37,39,36,38,28,40);(6,8)(9,12,14,30,16,29,15,24)(10,18,20,37,22,36,21,28)(11,13,25,32,27,31,26,33)(17,19,23,39,35,38,34,40);(6,8)(9,13)(10,19)(11,24)(12,25)(14,32)(15,33)(16,31)(17,28)(18,23)(20,39)(21,40)(22,38)(26,29)(27,30)(34,36)(35,37);(12,29)(13,31)(18,36)(19,38)(24,30)(28,37)(32,33)(39,40)

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=dhi,e^c=ei,e^d=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, 6350466);