#489608

order	512
nilpotency class	4
isoclinism	isoclinic class of #489390
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 ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

720 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)(3,11)(4,10)(5,8)(6,9)(7,13)(12,15)(14,16)(17,18)(19,31)(20,35)(21,29)(22,28)(23,30)(24,32)(25,33)(26,37)(27,38)(34,41)(36,46)(39,45)(40,44)(42,48)(43,47);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,16)(10,11)(17,19)(18,24)(20,36)(21,33)(22,40)(23,34)(25,26)(27,44)(28,42)(29,46)(30,43)(31,35)(32,38)(37,47)(39,48)(41,45);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,19)(18,25)(20,32)(21,40)(22,33)(23,34)(24,26)(27,43)(28,41)(29,31)(30,44)(35,46)(36,38)(37,48)(39,47)(42,45);(2,9)(3,12)(8,10)(13,14)(17,20,21,37,23,39,22,38)(18,26,28,35,30,27,29,45)(19,32,40,48,34,47,33,36)(24,41,46,44,43,31,42,25)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396804);