#6356458

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

1558 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)(19,27)(20,25)(21,23)(22,34)(24,30)(26,33)(28,41)(29,40)(31,37)(32,45)(35,36)(38,43)(39,44)(42,48)(46,47);(1,3,5,2,7,11,6,9)(4,10,13,8,12,16,14,15)(18,23)(24,36)(25,37)(26,34)(27,40)(38,47)(39,45)(41,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,12)(9,15)(11,16)(17,19,21,29)(18,24,23,36)(20,28,31,42)(22,35,33,30)(25,38,37,47)(26,40,34,27)(32,46,44,43)(39,48,45,41);(4,12)(8,15)(10,16)(13,14)(17,20)(18,25)(19,28)(21,31)(22,32)(23,37)(24,38)(26,39)(27,41)(29,42)(30,43)(33,44)(34,45)(35,46)(36,47)(40,48);(1,5,7,6)(2,3,9,11)(4,13,12,14)(8,10,15,16)(20,31)(25,37)(28,42)(32,44)(38,47)(39,45)(41,48)(43,46)

Polycyclic presentation:

2^9|b^2=fhi,c^2=i,e^2=h,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, 6263654);