#150958

order	512
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 200 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) = 72
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gh,e^2=fghi,g^2=hi,h^2=i,b^a=bd,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58154);