#6358921

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358842
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
prime radical	#18640	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1604 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)(3,7)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(17,18,23,28)(19,29,33,38)(20,25,22,27)(21,26,36,44)(24,32,40,30)(31,45,47,42)(34,41,37,43)(35,48,46,39);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19,23,33)(18,24,28,40)(20,30,22,32)(21,35,36,46)(25,38,27,29)(26,42,44,45)(31,37,47,34)(39,41,48,43);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(17,20,23,22)(18,25,28,27)(19,30,33,32)(21,34,36,37)(24,38,40,29)(26,41,44,43)(31,46,47,35)(39,42,48,45);(17,21)(18,26)(19,31)(20,34)(22,37)(23,36)(24,39)(25,41)(27,43)(28,44)(29,45)(30,46)(32,35)(33,47)(38,42)(40,48);(17,20,23,22)(18,27,28,25)(19,32,33,30)(21,34,36,37)(24,38,40,29)(26,43,44,41)(31,35,47,46)(39,42,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266117);