#6578116

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2019 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)(5,6,9,10)(7,11,8,12)(13,14,17,21)(15,23,26,19)(16,20,30,37)(18,22,31,38)(24,34,27,32)(25,39,41,33)(28,35,29,36)(40,44,42,43);(6,10)(11,12)(13,15,17,26)(14,19,21,23)(16,25,30,41)(18,24,31,27)(20,33,37,39)(22,32,38,34)(28,42,29,40)(35,44,36,43);(2,4)(15,24)(16,28)(18,31)(19,32)(20,35)(22,38)(23,34)(25,41)(26,27)(29,30)(33,39)(36,37);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,25)(17,30)(18,29)(19,33)(21,37)(22,36)(23,39)(24,40)(26,41)(27,42)(28,31)(32,43)(34,44)(35,38);(5,8,9,7)(6,12,10,11)(15,24)(16,29)(18,31)(19,32)(20,36)(22,38)(23,34)(26,27)(28,30)(35,37)(40,42)(43,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,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, 6485312);