#6729678

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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)(19,24)(20,25)(21,26)(22,28)(23,27)(29,37)(30,38)(31,43)(32,39)(33,40)(34,41)(35,42)(36,44)(45,47)(46,48);(1,3)(2,5)(4,8)(6,7)(17,19,23,32)(18,24,27,39)(20,33,35,45)(21,30,22,31)(25,40,42,47)(26,38,28,43)(29,36,46,34)(37,44,48,41);(10,14)(15,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,37)(26,41)(27,42)(28,44)(30,45)(31,33)(32,46)(38,47)(39,48)(40,43);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,22)(18,26,27,28)(19,30,32,31)(20,34,35,36)(24,38,39,43)(25,41,42,44)(29,45,46,33)(37,47,48,40);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,27)(19,31)(20,35)(21,22)(24,38)(29,45)(30,32)(33,46)(37,40)(39,43)(41,44)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=gi,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,e^c=ei,h^c=hi,e^d=ei,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, 6636874);