#6692608

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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

1424 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)(19,29,32,24)(20,25,23,28)(21,34,36,26)(30,42,37,38)(31,39,46,43)(33,41,35,40)(44,47,45,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(19,30)(20,23)(21,33)(24,38)(25,28)(26,40)(29,42)(32,37)(34,41)(35,36)(44,45)(47,48);(9,12,13,11)(10,14,15,16)(17,19,22,32)(18,24,27,29)(20,30,23,37)(21,31,36,46)(25,38,28,42)(26,39,34,43)(33,44,35,45)(40,47,41,48);(2,6)(5,7)(17,20,22,23)(18,25,27,28)(19,30,32,37)(21,35,36,33)(24,38,29,42)(26,41,34,40)(31,45,46,44)(39,48,43,47);(3,8)(5,7)(17,21,22,36)(18,26,27,34)(19,31,32,46)(20,33,23,35)(24,39,29,43)(25,40,28,41)(30,44,37,45)(38,47,42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6599804);