#6617402

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×#3518	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2916 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)(13,14,19,23)(15,24,27,34)(16,21,29,37)(17,22,18,31)(20,32,35,26)(25,39,42,44)(28,36,30,38)(33,40,43,41);(6,10)(8,11)(13,15,19,27)(14,20,23,35)(16,25,29,42)(17,26,18,32)(21,33,37,43)(22,34,31,24)(28,40,30,41)(36,39,38,44);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,21)(15,25)(17,30)(18,28)(19,29)(20,33)(22,38)(23,37)(24,39)(26,41)(27,42)(31,36)(32,40)(34,44)(35,43);(5,7,9,12)(6,11,10,8)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,28,29,30)(20,34,35,24)(21,36,37,38)(25,40,42,41)(33,39,43,44);(5,8,9,11)(6,7,10,12)(14,23)(20,35)(21,37)(22,31)(24,34)(33,43)(36,38)(39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,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, 6524598);