#6353728

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1608 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(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);(17,19,21,30,23,32,22,31)(18,24,36,43,27,38,26,28)(20,29,34,45,35,46,33,44)(25,37,39,47,41,48,40,42);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(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);(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)(2,8,9,13)(3,11,12,14)(7,15,16,10)(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=fh,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^c=dh,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260924);