#6467087

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

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=gi,h^2=i,b^a=bf,c^a=cg,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, 6374283);