#6714671

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

1415 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)(17,18,21,25)(19,27,30,23)(20,24,33,41)(22,26,35,43)(28,45,46,37)(29,39,31,38)(32,48,47,40)(34,42,36,44);(1,2)(3,6)(4,5)(7,8)(17,19,21,30)(18,23,25,27)(20,32,33,47)(22,29,35,31)(24,40,41,48)(26,38,43,39)(28,36,46,34)(37,44,45,42);(10,13)(14,16)(17,20)(18,24)(19,28)(21,33)(22,36)(23,37)(25,41)(26,44)(27,45)(29,32)(30,46)(31,47)(34,35)(38,40)(39,48)(42,43);(1,3,4,7)(2,6,5,8)(18,25)(20,33)(23,27)(26,43)(28,46)(32,47)(34,36)(38,39);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(19,29)(22,35)(23,38)(26,43)(27,39)(28,32)(30,31)(34,36)(37,40)(42,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6621867);