#150786

order	512
nilpotency class	5
isoclinism	isoclinic class of #150785
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

224 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 82 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)(4,7)(5,8)(10,15,12,14)(13,16)(18,25,27,45)(19,31)(20,37)(21,33)(22,39)(23,42)(26,30,47,44)(28,32,29,48)(34,36)(35,40)(38,41,46,43);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,18,24,27)(19,32,36,48)(20,38,35,46)(21,41,22,43)(23,44,42,30)(25,37,45,40)(26,31,47,34)(28,39,29,33);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,33,40,23,35,22,31,24,36,39,37,42,20,21,34)(18,26,41,38,30,45,29,48,27,47,43,46,44,25,28,32)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fh,d^2=gh,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57982);