#153020

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

228 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,7)(4,5)(6,8)(9,10,14,20,16,21,15,39)(11,22,27,52,29,42,28,63)(12,30,33,41,35,65,34,66)(13,36,37,48,38,67,31,19)(17,40,44,24,45,59,68,25)(18,23,46,54,47,55,64,32)(26,61,57,69,62,72,56,51)(43,60,50,53,70,58,49,71);(2,5)(6,7)(9,11,23,53,38,62,12,25,14,27,54,58,31,56,33,40,16,29,55,71,13,26,35,24,15,28,32,60,37,57,34,59)(10,17,41,61,48,70,47,42,39,68,30,51,36,50,46,52,21,45,66,72,19,43,18,22,20,44,65,69,67,49,64,63);(1,3,4,8)(2,6,5,7)(9,12,31,55,15,34,38,54,16,35,37,23,14,33,13,32)(10,18,36,41,20,46,48,65,21,47,67,66,39,64,19,30)(11,24,56,53,28,40,62,60,29,25,57,71,27,59,26,58)(17,42,50,72,44,63,70,51,45,22,49,61,68,52,43,69)

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=eh,d^2=gh,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,h^a=hi,c^b=ci

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, 60216);