#6358585

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358426
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

1559 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265781);