#6355437

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355407
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1597 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262633);