#489324

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

721 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)(17,18)(19,30)(20,26)(21,36)(22,28)(23,29)(24,32)(25,31)(27,38)(33,44)(34,45)(35,40)(37,41)(39,46)(42,48)(43,47);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,12)(10,16)(17,19)(18,24)(20,31)(21,37)(22,34)(23,33)(25,40)(26,41)(27,45)(28,42)(29,43)(30,36)(32,38)(35,47)(39,48)(44,46);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,16)(10,12)(17,19)(18,25)(20,34)(21,32)(22,31)(23,33)(24,40)(26,30)(27,43)(28,44)(29,45)(35,48)(36,41)(37,38)(39,47)(42,46);(1,4,6,5)(2,9,11,10)(3,13,15,14)(7,12,8,16)(17,20,23,22)(18,26,29,28)(19,31,33,34)(21,35,39,38)(24,41,43,42)(25,44,45,30)(27,36,40,46)(32,37,47,48)

Polycyclic presentation:

2^9|d^2=hi,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, 396520);