#6362719

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

1551 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)(4,12)(5,9)(6,10)(8,11)(13,15)(14,16)(17,18)(19,28)(20,25)(21,33)(22,36)(23,27)(24,29)(26,35)(30,43)(31,39)(32,44)(34,40)(37,41)(38,42)(45,47)(46,48);(1,3,6,5)(2,7,10,9)(4,11,13,14)(8,15,16,12)(17,19,23,31)(18,24,27,41)(20,29,22,37)(21,34,35,46)(25,39,36,28)(26,43,33,45)(30,38,47,32)(40,42,48,44);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,20,23,22)(18,25,27,36)(19,29,31,37)(21,32,35,38)(24,39,41,28)(26,42,33,44)(30,46,47,34)(40,43,48,45);(1,3,6,5)(2,7,10,9)(4,11,13,14)(8,15,16,12)(17,21,23,35)(18,26,27,33)(19,30,31,47)(20,32,22,38)(24,40,41,48)(25,42,36,44)(28,45,39,43)(29,46,37,34);(1,3,6,5)(2,9,10,7)(4,11,13,14)(8,12,16,15)(18,27)(24,41)(25,36)(26,33)(28,39)(40,48)(42,44)(43,45)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=hi,e^2=i,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, 6269915);