#6729900

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

1496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=g,e^2=g,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, 6637096);