#6467091

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2036 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374287);