#6617629

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
prime radical	C₂×#3524	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3024 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)(13,14,18,22,19,23,17,31)(15,24,26,35,27,34,32,20)(16,21,30,37,28,36,29,41)(25,38,40,43,39,44,42,33);(6,10)(8,11)(13,15,19,27)(14,20,23,35)(16,25,28,39)(17,26,18,32)(21,33,36,43)(22,34,31,24)(29,40,30,42)(37,44,41,38);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,21)(15,25)(17,29)(18,30)(19,28)(20,33)(22,37)(23,36)(24,38)(26,40)(27,39)(31,41)(32,42)(34,44)(35,43);(5,7,9,12)(6,11,10,8)(16,28)(21,36)(25,39)(29,30)(33,43)(37,41)(38,44)(40,42);(5,8,9,11)(6,7,10,12)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,29,28,30)(20,34,35,24)(21,37,36,41)(25,40,39,42)(33,44,43,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=hi,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, 6524825);