#489489

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
prime radical	#19355	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

651 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:

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

Polycyclic presentation:

2^9|b^2=hi,d^2=gi,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, 396685);