#6577935

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1952 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)(8,12)(9,10)(13,14,19,16)(15,20,27,28)(17,21,35,29)(18,22,36,30)(23,25,38,26)(24,37,31,39)(32,34,40,33)(41,42,44,43);(6,10)(11,12)(13,15,19,27)(14,20,16,28)(17,31,35,24)(18,25,36,26)(21,39,29,37)(22,38,30,23)(32,41,40,44)(33,43,34,42);(2,4)(13,16,19,14)(15,23,27,38)(17,32,35,40)(18,22,36,30)(20,25,28,26)(21,34,29,33)(24,37,31,39)(41,43,44,42);(5,7,9,8)(6,11,10,12)(13,17,19,35)(14,21,16,29)(15,24,27,31)(18,33,36,34)(20,37,28,39)(22,32,30,40)(23,41,38,44)(25,42,26,43);(5,8,9,7)(6,12,10,11)(15,25)(17,33)(18,36)(20,38)(21,32)(22,30)(23,28)(26,27)(29,40)(34,35)(41,44)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,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, 6485131);