#489323

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|d^2=h,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fhi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396519);