#489613

order	512
nilpotency class	4
isoclinism	isoclinic class of #489398
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 ◁ C₂×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

738 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)(19,28)(20,32)(21,37)(22,27)(23,29)(24,42)(25,36)(26,35)(30,39)(31,43)(33,46)(34,44)(38,45)(40,47)(41,48);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,16)(10,11)(17,19)(18,23)(20,33)(21,38)(22,30)(24,43)(25,45)(26,46)(27,40)(28,32)(29,36)(31,35)(34,48)(37,41)(39,44)(42,47);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(20,34)(24,25)(28,39)(29,47)(32,44)(33,48)(36,42)(43,45);(2,9)(3,12)(8,10)(13,14)(17,20,35,42,22,34,21,36)(18,24,37,32,27,25,26,44)(19,29,38,48,30,47,31,33)(23,39,46,45,40,28,41,43)

Polycyclic presentation:

2^9|d^2=gh,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,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, 396809);