#6624657

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1499 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)(17,18,21,27,23,26,22,28)(19,29,31,24,33,40,32,39)(20,34,36,41,37,42,35,25)(30,44,46,48,47,38,45,43);(1,3,5,4)(2,6,7,8)(10,14)(12,15)(17,19,23,33)(18,24,26,39)(20,30,37,47)(21,32,22,31)(25,38,41,44)(27,29,28,40)(34,48,42,43)(35,46,36,45);(9,10)(11,12)(13,14)(15,16)(17,20,23,37)(18,25,26,41)(19,30,33,47)(21,35,22,36)(24,38,39,44)(27,42,28,34)(29,43,40,48)(31,45,32,46);(1,4,5,3)(2,6,7,8)(9,11,13,16)(10,15,14,12)(18,26)(24,39)(25,41)(27,28)(29,40)(34,42)(38,44)(43,48);(9,12,13,15)(10,11,14,16)(17,21,23,22)(18,27,26,28)(19,31,33,32)(20,35,37,36)(24,40,39,29)(25,42,41,34)(30,45,47,46)(38,48,44,43)

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=i,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,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, 6531853);