#489653

order	512
nilpotency class	4
isoclinism	isoclinic class of #489365
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	C₂³	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

646 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:

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

Polycyclic presentation:

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

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, 396849);