#486736

order	512
nilpotency class	4
isoclinism	isoclinic class of #486327
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	#28174	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

719 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 719 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 128

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

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

Polycyclic presentation:

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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393932);