#6543974

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	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1417 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=fg,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,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, 6451170);