#6366149

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365694
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1534 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)(3,8)(4,13)(5,11)(6,15)(7,10)(9,14)(12,16)(17,18,22,25)(19,27,30,37)(20,24,33,39)(21,34,35,26)(23,29,38,31)(28,41,43,47)(32,45,46,40)(36,42,48,44);(1,3,7,4)(2,8,10,13)(5,14,6,12)(9,15,16,11)(17,19,22,30)(18,23,25,38)(20,28,33,43)(21,29,35,31)(24,36,39,48)(26,37,34,27)(32,42,46,44)(40,47,45,41);(1,4,7,3)(2,9,10,16)(5,14,6,12)(8,15,13,11)(17,20,22,33)(18,24,25,39)(19,28,30,43)(21,32,35,46)(23,36,38,48)(26,40,34,45)(27,41,37,47)(29,42,31,44);(2,10)(8,13)(9,16)(11,15)(19,29)(21,35)(23,37)(26,34)(27,38)(28,42)(30,31)(32,46)(36,47)(40,45)(41,48)(43,44);(1,5,7,6)(2,11,10,15)(3,12,4,14)(8,16,13,9)(18,25)(19,30)(24,39)(26,34)(28,43)(29,31)(40,45)(42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273345);