#6578563

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	C₂×#3341	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1998 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)(13,14,19,16)(15,20,26,27)(17,21,34,28)(18,22,36,29)(23,35,40,25)(24,37,30,38)(31,32,39,33)(41,42,44,43);(5,7,9,8)(6,10,12,11)(13,15,19,26)(14,20,16,27)(17,30,34,24)(18,35,36,25)(21,38,28,37)(22,40,29,23)(31,44,39,41)(32,43,33,42);(2,4)(13,16,19,14)(15,23,26,40)(17,31,34,39)(18,22,36,29)(20,35,27,25)(21,32,28,33)(24,38,30,37)(41,42,44,43);(5,7,9,8)(6,11,12,10)(13,17)(14,21)(15,24)(16,28)(18,32)(19,34)(20,37)(22,39)(23,41)(25,43)(26,30)(27,38)(29,31)(33,36)(35,42)(40,44);(5,8,9,7)(6,10,12,11)(15,25)(17,32)(18,36)(20,23)(21,39)(22,29)(24,30)(26,35)(27,40)(28,31)(33,34)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485759);