#130819

order	512
nilpotency class	4
isoclinism	isoclinic class of #130738
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

559 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=ei,f^c=fg

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38015);