#6728191

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1251 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)(3,6)(4,7)(5,8)(9,10,13,15)(11,14,12,16)(17,18)(19,27)(20,31)(21,25)(22,26)(23,30)(24,33)(28,38)(29,39)(32,41)(34,44)(35,43)(36,40)(37,42)(45,46)(47,48);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,14)(13,15)(17,19,21,30)(18,23,25,27)(20,32,33,47)(22,36,35,29)(24,41,31,48)(26,39,43,40)(28,37,46,34)(38,44,45,42);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,14,15,16)(17,20)(18,24)(19,28)(21,33)(22,37)(23,38)(25,31)(26,44)(27,45)(29,47)(30,46)(32,36)(34,35)(39,41)(40,48)(42,43);(9,12,13,11)(10,14,15,16)(18,25)(20,33)(23,27)(26,43)(28,46)(32,47)(34,37)(39,40);(1,4,5,3)(2,7,8,6)(18,25)(19,29)(22,35)(23,39)(24,31)(27,40)(28,47)(30,36)(32,46)(34,37)(38,41)(45,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=fg,d^2=fi,e^2=gi,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,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6635387);