#489269

order	512
nilpotency class	4
isoclinism	isoclinic class of #488895
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

717 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396465);