#6442831

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2602 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,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, 6350027);