#6358967

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	#62102	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1549 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^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,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, 6266163);