#6577517

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1969 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)(7,11)(8,10)(9,12)(13,14,18,23)(15,25,28,20)(16,30,33,21)(17,34,36,22)(19,24,37,32)(26,39,43,35)(27,40,44,31)(29,42,38,41);(5,7,9,8)(6,10,12,11)(13,15,18,28)(14,20,23,25)(16,31,33,40)(17,35,36,39)(19,38,37,29)(21,27,30,44)(22,26,34,43)(24,42,32,41);(2,4)(5,7,9,8)(6,11,12,10)(13,16,18,33)(14,21,23,30)(15,26,28,43)(17,32,36,24)(19,34,37,22)(20,39,25,35)(27,41,44,42)(29,40,38,31);(5,8,9,7)(6,10,12,11)(16,22)(17,30)(19,37)(21,36)(24,32)(26,31)(27,39)(29,38)(33,34)(35,44)(40,43)(41,42);(13,17,18,36)(14,22,23,34)(15,27,28,44)(16,32,33,24)(19,30,37,21)(20,40,25,31)(26,41,43,42)(29,39,38,35)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^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, 6484713);