#6686596

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
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
prime radical	#61294	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1582 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)(3,10)(4,13)(5,8)(6,9)(7,12)(11,15)(14,16)(17,18)(19,24)(20,33)(21,26)(22,38)(23,27)(25,35)(28,42)(29,30)(31,40)(32,41)(34,47)(36,43)(37,44)(39,45)(46,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,24)(20,34)(21,29)(22,31)(23,32)(25,42)(26,30)(27,41)(28,35)(33,47)(36,45)(37,46)(38,40)(39,43)(44,48);(17,20,22,35,23,36,21,37)(18,25,26,33,27,44,38,43)(19,28,31,45,32,46,29,34)(24,39,30,42,41,47,40,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,21,23,22)(18,26,27,38)(19,29,32,31)(20,35,36,37)(24,30,41,40)(25,43,44,33)(28,45,46,34)(39,48,47,42);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,18)(19,30)(20,25)(21,38)(22,26)(23,27)(24,29)(28,42)(31,41)(32,40)(33,35)(34,39)(36,44)(37,43)(45,47)(46,48)

Polycyclic presentation:

2^9|c^2=g,d^2=fi,e^2=f,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,d^c=di,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, 6593792);