#135617

order	512
nilpotency class	4
isoclinism	isoclinic class of #135598
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1823
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

308 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=d,e^2=gh,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=ehi,f^c=fgi,h^c=hi

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, 42813);