#6692724

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1462 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)(17,18,21,25)(19,27,31,23)(20,24,33,40)(22,26,35,42)(28,38,30,36)(29,39,32,37)(34,41,45,48)(43,47,44,46);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,32)(21,31)(22,28)(24,39)(25,27)(26,36)(29,33)(30,35)(34,44)(37,40)(38,42)(41,47)(43,45)(46,48);(9,12,13,11)(10,14,15,16)(19,28)(22,35)(23,36)(26,42)(27,38)(29,43)(30,31)(32,44)(34,45)(37,46)(39,47)(41,48);(2,6)(5,7)(17,20)(18,24)(19,29)(21,33)(22,34)(23,37)(25,40)(26,41)(27,39)(28,43)(30,44)(31,32)(35,45)(36,46)(38,47)(42,48);(3,8)(5,7)(19,30)(20,33)(22,35)(23,38)(24,40)(26,42)(27,36)(28,31)(29,43)(32,44)(37,46)(39,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,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, 6599920);