#6744258

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1301 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)(3,7)(4,8)(5,6)(9,10,12,13)(11,15,16,14)(17,18)(19,28)(20,33)(21,37)(22,25)(23,27)(24,31)(26,34)(29,38)(30,39)(32,41)(35,40)(36,46)(42,45)(43,48)(44,47);(10,13)(11,16)(17,19,22,31)(18,24,25,28)(20,30,34,45)(21,32,23,29)(26,39,33,42)(27,41,37,38)(35,44,36,43)(40,48,46,47);(2,6)(7,8)(9,11,12,16)(10,14,13,15)(18,25)(19,29)(20,34)(21,23)(24,38)(28,41)(30,44)(31,32)(39,48)(40,46)(42,47)(43,45);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,22,34)(18,26,25,33)(19,30,31,45)(21,35,23,36)(24,39,28,42)(27,40,37,46)(29,43,32,44)(38,47,41,48);(1,4,5,3)(2,8,6,7)(17,21)(18,27)(19,31)(20,35)(22,23)(25,37)(26,40)(30,45)(33,46)(34,36)(38,41)(47,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fi,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,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, 6651454);