#6563585

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6470781);