#6362593

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362510
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

1514 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)(19,29,32,43)(20,33,36,25)(21,27,22,26)(24,37,40,31)(30,41,47,45)(34,39,46,48)(35,42,38,44);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19,23,32)(18,24,28,40)(20,34,36,46)(21,31,22,37)(25,41,33,45)(26,29,27,43)(30,38,47,35)(39,44,48,42);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11);(17,20,23,36)(18,25,28,33)(19,30,32,47)(21,35,22,38)(24,39,40,48)(26,42,27,44)(29,45,43,41)(31,46,37,34);(2,9)(7,13)(8,16)(11,15)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,35,36,38)(24,29,40,43)(25,42,33,44)(30,46,47,34)(39,45,48,41)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269789);