#486703

order	512
nilpotency class	4
isoclinism	isoclinic class of #486291
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 ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

791 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 791 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265737
maximal normal subgroup of	#72265737

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393899);