#6367720

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 6274916);