#6730515

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1448 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=gi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6637711);