#6367719

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367534
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

1511 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274915);