#6714678

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6621874);