#6674245

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|c^2=f,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei,g^c=gi,h^c=hi,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, 6581441);