#488707

order	512
nilpotency class	4
isoclinism	isoclinic class of #488707
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	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

730 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)(3,6)(4,7)(5,8)(9,15)(10,13)(11,14)(12,16)(18,24)(19,29)(20,21)(22,38)(25,42)(26,27)(28,41)(30,47)(32,33)(34,35)(36,39)(40,44)(43,45)(46,48);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,18)(19,25)(20,24)(21,36)(22,39)(23,28)(26,34)(27,38)(29,45)(30,40)(31,42)(32,44)(33,46)(35,41)(37,48)(43,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,19)(18,25)(20,30)(21,37)(22,32)(23,33)(24,40)(26,42)(27,45)(28,46)(29,38)(31,34)(35,47)(36,48)(39,44)(41,43);(2,8)(6,13)(7,14)(12,15)(17,20,22,34,23,35,38,21)(18,26,27,24,28,36,39,41)(19,30,32,31,33,47,29,37)(25,42,45,40,46,48,44,43)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395903);