#6350595

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350525
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2749 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)(19,30)(20,35)(21,27)(22,28)(23,29)(24,33)(25,36)(26,34)(31,45)(32,40)(37,43)(38,42)(39,48)(41,46)(44,47);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19)(18,24)(20,31)(21,32)(22,33)(23,34)(25,39)(26,28)(27,30)(29,40)(35,41)(36,46)(37,47)(38,48)(42,44)(43,45);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,38)(18,25,29,43)(19,31,34,48)(21,36,22,37)(24,39,40,45)(26,41,30,44)(27,42,28,35)(32,46,33,47);(2,9)(7,16)(8,13)(10,15)(17,19)(18,26)(20,31)(21,33)(22,32)(23,34)(24,28)(25,41)(27,40)(29,30)(35,39)(36,47)(37,46)(38,48)(42,45)(43,44);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,22)(18,27,29,28)(19,32,34,33)(20,36,38,37)(24,30,40,26)(25,42,43,35)(31,46,48,47)(39,44,45,41)

Polycyclic presentation:

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

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, 6257791);