#6450082

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6,8,9)(7,10,11,12)(13,14,19,23)(15,24,29,42)(16,21,31,39)(17,22,18,33)(20,27,36,28)(25,40,26,41)(30,37,32,38)(34,44,35,43);(1,2)(3,4)(13,15,18,28,19,29,17,27)(14,20,22,24,23,36,33,42)(16,25,30,43,31,26,32,44)(21,34,38,40,39,35,37,41);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,32)(18,30)(19,31)(20,34)(22,38)(23,39)(24,40)(26,29)(27,44)(28,43)(33,37)(35,36)(41,42);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,26)(17,32)(18,30)(19,31)(20,35)(22,38)(23,39)(24,41)(25,29)(27,43)(28,44)(33,37)(34,36)(40,42);(13,17,19,18)(14,22,23,33)(15,27,29,28)(16,30,31,32)(20,24,36,42)(21,37,39,38)(25,43,26,44)(34,41,35,40)

Polycyclic presentation:

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

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, 6357278);