#6559684

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1533 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)(17,18,21,25)(19,27,29,23)(20,31,33,46)(22,36,35,26)(24,39,41,34)(28,43,32,47)(30,38,45,42)(37,44,40,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,32)(21,29)(22,30)(24,40)(25,27)(26,38)(28,33)(31,43)(34,48)(35,45)(36,42)(37,41)(39,44)(46,47);(10,13)(14,16)(17,20,22,34,21,33,35,39)(18,24,26,31,25,41,36,46)(19,28,30,44,29,32,45,48)(23,37,38,47,27,40,42,43);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(19,29)(20,33)(23,27)(24,41)(30,45)(31,46)(34,39)(38,42);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(19,29)(23,27)(28,32)(30,45)(37,40)(38,42)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6466880);