#6444243

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	C₂×#3060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 6351439);