#6444204

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#3154	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2767 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)(11,21)(12,18)(13,19)(14,17)(15,20)(16,24)(22,30)(23,37)(25,32)(26,40)(27,35)(28,33)(29,36)(31,38)(34,39);(6,7)(9,11,14,24)(10,16,17,21)(12,26,28,39)(13,25,15,22)(18,34,33,40)(19,30,20,32)(23,29,38,27)(31,36,37,35);(1,2,3,4)(5,6,8,7)(10,17)(11,22)(13,15)(16,30)(18,33)(21,32)(23,26)(24,25)(27,29)(31,40)(34,37)(38,39);(5,7,8,6)(9,12,14,28)(10,18,17,33)(11,23,24,38)(13,27,15,29)(16,31,21,37)(19,35,20,36)(22,26,25,39)(30,40,32,34);(9,13,14,15)(10,19,17,20)(11,22,24,25)(12,27,28,29)(16,32,21,30)(18,35,33,36)(23,26,38,39)(31,34,37,40)

Polycyclic presentation:

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

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, 6351400);