#135596

order	512
nilpotency class	4
isoclinism	isoclinic class of #135590
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	C₂³×C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

326 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,10,12,6)(4,9)(5,8)(7,11,14,16)(13,15)(17,18,21,28,22,26,23,38)(19,29,31,42,32,24,37,41)(20,33,35,43,36,25,34,27)(30,39,46,45,47,44,48,40);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,16)(14,15)(17,19,22,32)(18,24,26,29)(20,30,36,47)(21,37,23,31)(25,39,33,44)(27,40,43,45)(28,42,38,41)(34,46,35,48);(1,3)(2,7,8,14)(4,11)(5,12)(6,9,10,15)(13,16)(18,25,26,33)(20,34)(21,23)(24,39,29,44)(27,28,43,38)(30,46)(31,37)(35,36)(40,42,45,41)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42792);