#6357706

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357668
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1834 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)(3,12)(4,15)(5,10)(6,16)(7,11)(8,9)(13,14)(17,18)(19,28)(20,25)(21,35)(22,27)(23,31)(24,33)(26,29)(30,42)(32,43)(34,39)(36,48)(37,47)(38,44)(40,45)(41,46);(1,3,5,14,7,15,6,9)(2,8,16,4,11,13,10,12)(17,19,29,44,22,32,21,31)(18,23,35,43,27,38,26,28)(20,30,45,48,33,46,34,47)(24,36,40,42,25,37,39,41);(1,4,5,13,7,12,6,8)(2,9,16,3,11,14,10,15)(17,20)(18,24)(19,30)(21,34)(22,33)(23,36)(25,27)(26,39)(28,41)(29,45)(31,47)(32,46)(35,40)(37,38)(42,43)(44,48);(1,2)(3,13)(4,9)(5,16)(6,10)(7,11)(8,15)(12,14)(17,20)(18,25)(19,30)(21,34)(22,33)(23,37)(24,27)(26,40)(28,42)(29,45)(31,47)(32,46)(35,39)(36,38)(41,43)(44,48);(1,5,7,6)(2,10,11,16)(3,14,15,9)(4,8,12,13)(20,33)(24,25)(30,46)(34,45)(36,37)(39,40)(41,42)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264902);