#6687396

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 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)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18,23,22)(19,28,32,24)(20,34,39,44)(21,27,36,26)(25,37,38,40)(29,43,47,48)(30,42,33,31)(35,45,46,41);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,19)(18,24)(20,35)(21,30)(22,28)(23,32)(25,43)(26,42)(27,31)(29,37)(33,36)(34,41)(38,48)(39,46)(40,47)(44,45);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,36,37,23,39,21,40)(18,25,27,44,22,38,26,34)(19,29,33,46,32,47,30,35)(24,41,31,43,28,45,42,48);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,16,11,15)(17,21,23,36)(18,26,22,27)(19,30,32,33)(20,37,39,40)(24,42,28,31)(25,44,38,34)(29,46,47,35)(41,43,45,48);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,22,23,18)(19,31,32,42)(20,38,39,25)(21,27,36,26)(24,33,28,30)(29,43,47,48)(34,40,44,37)(35,41,46,45)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=f,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,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, 6594592);