#6356512

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356405
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1576 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)(3,5)(4,8)(6,11)(7,9)(10,13)(12,15)(14,16)(17,18)(19,28)(20,26)(21,27)(22,24)(23,35)(25,30)(29,43)(31,40)(32,41)(33,38)(34,42)(36,37)(39,46)(44,47)(45,48);(1,3,5,2,7,11,6,9)(4,10,13,8,12,16,14,15)(18,24)(25,37)(26,38)(27,35)(28,40)(39,47)(41,42)(43,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,12)(9,15)(11,16)(17,19)(18,25)(20,29)(21,30)(22,31)(23,36)(24,37)(26,39)(27,40)(28,35)(32,44)(33,45)(34,46)(38,47)(41,43)(42,48);(4,12)(8,15)(10,16)(13,14)(17,20,22,33)(18,26,24,38)(19,29,31,45)(21,34,23,32)(25,39,37,47)(27,42,35,41)(28,43,40,48)(30,46,36,44);(1,5,7,6)(2,3,9,11)(4,13,12,14)(8,10,15,16)(17,21,22,23)(18,27,24,35)(19,30,31,36)(20,32,33,34)(25,40,37,28)(26,41,38,42)(29,44,45,46)(39,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263708);