#488723

order	512
nilpotency class	4
isoclinism	isoclinic class of #488723
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

757 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)(17,18)(19,32)(20,37)(21,42)(22,30)(23,29)(24,31)(25,34)(26,35)(27,40)(28,41)(33,47)(36,48)(38,46)(39,45)(43,44);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19)(18,25)(20,38)(21,33)(22,35)(23,43)(24,36)(26,37)(27,48)(28,44)(29,45)(30,47)(31,46)(32,42)(34,41)(39,40);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,19)(18,26)(20,39)(21,34)(22,43)(23,35)(24,36)(25,37)(27,45)(28,46)(29,48)(30,32)(31,44)(33,41)(38,40)(42,47);(2,8)(6,13)(7,14)(12,15)(17,20,23,21,24,41,22,40)(18,27,30,28,31,42,29,37)(19,33,35,38,36,39,43,34)(25,44,45,48,46,26,47,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395919);