#6366664

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366542
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1501 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)(3,7)(4,10)(5,14)(6,9)(8,11)(12,15)(13,16)(18,24)(21,22)(25,37)(26,38)(27,28)(30,35)(32,41)(33,36)(39,47)(40,42)(43,48)(44,45);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,18,23,28)(19,25,31,40)(20,32,34,38)(21,24,22,27)(26,36,41,33)(29,43,46,47)(30,42,35,37)(39,44,48,45);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,19)(18,25)(20,29)(21,35)(22,30)(23,31)(24,37)(26,39)(27,42)(28,40)(32,43)(33,45)(34,46)(36,44)(38,47)(41,48);(2,9)(7,15)(8,16)(10,14)(17,20,23,34)(18,26,28,41)(19,29,31,46)(21,33,22,36)(24,38,27,32)(25,39,40,48)(30,44,35,45)(37,47,42,43);(1,4,6,5)(2,10,9,14)(3,11,12,13)(7,8,15,16)(17,21,23,22)(18,27,28,24)(19,30,31,35)(20,33,34,36)(25,37,40,42)(26,32,41,38)(29,44,46,45)(39,47,48,43)

Polycyclic presentation:

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

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, 6273860);