#6349427

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349370
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2686 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256623);