#6687401

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

1670 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)(17,18,24,27)(19,25,32,28)(20,23,35,21)(22,26,39,44)(29,33,31,41)(30,40,47,45)(34,38,43,36)(37,42,46,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,19)(18,25)(20,33)(21,29)(22,37)(23,31)(24,32)(26,42)(27,28)(30,38)(34,45)(35,41)(36,47)(39,46)(40,43)(44,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,23,27,24,35,21,18)(19,28,31,41,32,25,29,33)(22,34,38,44,39,43,36,26)(30,45,46,48,47,40,37,42);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,16,11,15)(17,21,24,23)(18,20,27,35)(19,29,32,31)(22,36,39,38)(25,33,28,41)(26,34,44,43)(30,37,47,46)(40,42,45,48);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,22,24,39)(18,26,27,44)(19,30,32,47)(20,34,35,43)(21,36,23,38)(25,40,28,45)(29,37,31,46)(33,42,41,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fi,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, 6594597);