#6578478

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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1976 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);(5,7,9,8)(6,10,12,11)(13,14)(15,24)(16,20)(17,21)(18,23)(19,26)(22,33)(25,41)(27,39)(28,44)(29,37)(30,40)(31,38)(32,35)(34,42)(36,43);(2,4)(13,15,18,27)(14,19,23,35)(16,28,31,42)(17,32,33,26)(20,36,38,41)(21,24,22,39)(25,29,43,30)(34,40,44,37);(5,7,9,8)(6,11,12,10)(13,16,18,31)(14,20,23,38)(15,25,27,43)(17,29,33,30)(19,34,35,44)(21,37,22,40)(24,41,39,36)(26,42,32,28);(5,8,9,7)(6,10,12,11)(14,21)(16,29)(17,33)(19,24)(22,23)(25,42)(26,32)(28,43)(30,31)(34,44)(35,39)(37,40)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=fi,e^2=f,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, 6485674);