#6543789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1458 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6450985);