#6693859

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

1299 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)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,31)(28,35)(29,41)(30,40)(32,37)(33,39)(34,38)(36,44)(42,46)(43,47)(45,48);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,29)(21,31)(22,28)(24,27)(25,36)(26,40)(30,34)(32,44)(33,42)(35,38)(37,41)(39,47)(43,45)(46,48);(9,12,13,11)(10,14,15,16)(18,24)(19,28)(20,32)(22,34)(23,35)(27,40)(29,43)(30,31)(36,47)(39,48)(41,46)(42,44);(2,5)(6,8)(17,20,21,32)(18,25,24,37)(19,29,31,44)(22,33,34,45)(23,36,27,41)(26,39,38,48)(28,42,30,43)(35,46,40,47);(1,3,4,7)(2,6,5,8)(19,30)(22,34)(23,35)(26,38)(27,40)(28,31)(29,43)(33,45)(36,46)(39,48)(41,47)(42,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,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, 6601055);