#6728227

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1256 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)(20,32)(21,25)(22,26)(23,30)(24,34)(28,38)(29,39)(31,43)(33,40)(35,44)(36,42)(37,41)(45,46)(47,48);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,14)(13,15)(17,19)(18,23)(20,33)(21,30)(22,29)(24,40)(25,27)(26,43)(28,35)(31,36)(32,48)(34,47)(37,46)(38,41)(39,42)(44,45);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,14,15,16)(17,20)(18,24)(19,28)(21,34)(22,37)(23,38)(25,32)(26,44)(27,45)(29,33)(30,46)(31,47)(35,36)(39,48)(40,43)(41,42);(9,12,13,11)(10,14,15,16)(18,25)(20,34)(23,27)(26,42)(28,46)(33,47)(35,37)(39,43);(1,4,5,3)(2,7,8,6)(18,25)(19,29)(22,36)(23,39)(24,32)(27,43)(28,33)(30,31)(35,37)(38,48)(40,45)(46,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,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=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, 6635423);