#6773356

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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

1463 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)(19,23)(20,25)(21,33)(22,27)(24,32)(26,35)(28,37)(29,38)(30,45)(31,41)(34,40)(36,42)(39,44)(43,48)(46,47);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,23,27,38)(20,28,31,43)(21,30,35,44)(24,36,40,47)(25,37,41,48)(26,39,33,45)(32,42,34,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,24)(19,28)(21,34)(22,31)(23,36)(25,33)(26,41)(27,40)(29,43)(30,46)(32,35)(37,45)(38,47)(39,48)(42,44);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,28)(21,32)(22,31)(23,37)(24,33)(26,40)(27,41)(29,43)(30,42)(34,35)(36,45)(38,48)(39,47)(44,46);(10,13)(14,16)(19,29)(20,31)(23,38)(24,40)(25,41)(30,44)(32,34)(39,45)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6680552);