#6444188

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

2769 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)(9,10,13,17)(11,15,21,31)(12,23,25,16)(14,18,27,35)(19,37,38,29)(20,30,22,32)(24,40,39,33)(26,36,28,34);(6,7)(9,11,13,21)(10,15,17,31)(12,24,25,39)(14,20,27,22)(16,33,23,40)(18,30,35,32)(19,28,38,26)(29,36,37,34);(1,2,3,4)(5,6,8,7)(9,12,13,25)(10,16,17,23)(11,19,21,38)(14,28,27,26)(15,29,31,37)(18,36,35,34)(20,24,22,39)(30,33,32,40);(5,7,8,6)(11,20)(14,27)(15,30)(18,35)(19,24)(21,22)(26,28)(29,33)(31,32)(34,36)(37,40)(38,39);(12,25)(16,23)(19,38)(24,39)(26,28)(29,37)(33,40)(34,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,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, 6351384);