#6456974

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1432 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)(17,18,21,25)(19,23,29,39)(20,31,33,24)(22,26,35,43)(27,45,46,37)(28,38,30,40)(32,48,47,41)(34,44,36,42);(1,3,5,4)(2,6,8,7)(17,19)(18,23)(20,32)(21,29)(22,28)(24,41)(25,39)(26,38)(27,34)(30,35)(31,48)(33,47)(36,46)(37,42)(40,43)(44,45);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20)(18,24)(19,27)(21,33)(22,36)(23,37)(25,31)(26,44)(28,32)(29,46)(30,47)(34,35)(38,41)(39,45)(40,48)(42,43);(1,4,5,3)(2,6,8,7)(19,28)(22,35)(23,38)(26,43)(27,32)(29,30)(34,36)(37,41)(39,40)(42,44)(45,48)(46,47);(9,11,12,15)(10,14,13,16)(20,33)(24,31)(27,46)(32,47)(34,36)(37,45)(41,48)(42,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,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, 6364170);