#6456932

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,25)(19,23,29,39)(20,24,33,41)(21,26,35,43)(27,36,45,47)(28,37,30,38)(31,40,46,48)(32,42,34,44);(1,3,5,4)(2,6,8,7)(17,19)(18,23)(20,31)(21,28)(22,29)(24,40)(25,39)(26,37)(27,32)(30,35)(33,46)(34,45)(36,42)(38,43)(41,48)(44,47);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,22,33)(18,24,25,41)(19,27,29,45)(21,34,35,32)(23,36,39,47)(26,44,43,42)(28,31,30,46)(37,40,38,48);(1,4,5,3)(2,6,8,7)(19,28)(21,35)(23,37)(26,43)(27,31)(29,30)(32,34)(36,40)(38,39)(42,44)(45,46)(47,48);(9,11,12,15)(10,14,13,16)(18,25)(19,29)(24,41)(26,43)(27,45)(28,30)(31,46)(42,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fi,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=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, 6364128);