#6718225

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

1486 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18,22,27)(19,28,32,24)(20,33,36,25)(21,26,39,23)(29,42,48,45)(30,31,38,40)(34,43,47,46)(35,41,44,37);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,34)(21,38)(22,32)(23,40)(25,43)(26,31)(27,28)(29,37)(30,39)(33,46)(35,45)(36,47)(41,48)(42,44);(10,15)(14,16)(17,20,22,36)(18,25,27,33)(19,29,32,48)(21,35,39,44)(23,41,26,37)(24,42,28,45)(30,46,38,43)(31,47,40,34);(2,6)(5,7)(17,21)(18,26)(19,30)(20,35)(22,39)(23,27)(24,40)(25,37)(28,31)(29,46)(32,38)(33,41)(34,42)(36,44)(43,48)(45,47);(3,8)(5,7)(9,12,13,11)(10,16,15,14)(19,31)(21,39)(23,26)(24,30)(28,38)(29,47)(32,40)(34,48)(35,44)(37,41)(42,46)(43,45)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,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, 6625421);