#6576869

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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
prime radical	#35687	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2672 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)(7,12)(8,10)(9,11)(13,14,18,23)(15,25,28,20)(16,30,33,21)(17,34,35,22)(19,24,32,41)(26,37,44,43)(27,38,31,40)(29,42,36,39);(5,7,9,8)(6,10,11,12)(13,15,18,28)(14,20,23,25)(16,31,33,27)(17,26,35,44)(19,36,32,29)(21,40,30,38)(22,37,34,43)(24,42,41,39);(2,4)(13,16)(14,21)(15,26)(17,32)(18,33)(19,35)(20,37)(22,41)(23,30)(24,34)(25,43)(27,29)(28,44)(31,36)(38,39)(40,42);(5,8,9,7)(6,10,11,12)(16,17)(19,32)(21,22)(24,41)(26,31)(27,44)(29,36)(30,34)(33,35)(37,40)(38,43)(39,42);(13,17,18,35)(14,22,23,34)(15,27,28,31)(16,32,33,19)(20,38,25,40)(21,41,30,24)(26,29,44,36)(37,39,43,42)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6484065);