#6617630

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3019 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)(13,14,18,24,19,23,17,22)(15,25,27,34,28,38,32,20)(16,21,29,35,30,36,31,37)(26,39,41,44,42,43,40,33);(6,10)(8,11)(13,15,19,28)(14,20,23,34)(16,26,30,42)(17,27,18,32)(21,33,36,44)(22,25,24,38)(29,40,31,41)(35,43,37,39);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,21)(15,26)(17,31)(18,29)(19,30)(20,33)(22,37)(23,36)(24,35)(25,39)(27,41)(28,42)(32,40)(34,44)(38,43);(5,7,9,12)(6,11,10,8)(13,17,19,18)(14,22,23,24)(15,27,28,32)(16,29,30,31)(20,25,34,38)(21,35,36,37)(26,40,42,41)(33,43,44,39);(5,8,9,11)(6,7,10,12)(14,23)(20,34)(21,36)(22,24)(25,38)(33,44)(35,37)(39,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524826);