#6687397

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

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=f,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,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, 6594593);