#488786

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
prime radical	#62341	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 675 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) = 256

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

(1,2,5,8)(3,9,10,16)(4,12,14,7)(6,13,15,11)(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)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,18,23,28)(19,25,33,46)(20,24,35,41)(21,36,34,26)(22,39,38,27)(29,45,32,44)(30,40,47,43)(31,42,37,48);(4,14)(7,12)(9,16)(11,13)(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);(3,10)(6,15)(9,16)(11,13)(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|a^2=hi,b^2=i,d^2=ghi,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, 395982);