#6444164

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2750 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)(5,6,8,7)(9,10)(11,22)(12,18)(13,19)(14,30)(15,21)(16,20)(17,25)(23,39)(24,33)(26,38)(27,37)(28,35)(29,36)(31,34)(32,40);(6,7)(9,11)(10,17)(12,26)(13,28)(14,24)(15,25)(16,31)(18,35)(19,38)(20,33)(21,22)(23,27)(29,40)(30,34)(32,37)(36,39);(1,2,3,4)(5,6,8,7)(9,12,15,13)(10,18,21,19)(11,23,25,40)(14,27,16,29)(17,32,22,39)(20,36,30,37)(24,28,31,26)(33,35,34,38);(5,7,8,6)(9,13,15,12)(10,19,21,18)(11,23,25,40)(14,29,16,27)(17,32,22,39)(20,37,30,36)(24,28,31,26)(33,35,34,38);(9,14,15,16)(10,20,21,30)(11,24,25,31)(12,27,13,29)(17,33,22,34)(18,36,19,37)(23,28,40,26)(32,35,39,38)

Polycyclic presentation:

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

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, 6351360);