#150876

order	512
nilpotency class	5
isoclinism	isoclinic class of #150788
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	C₂⁴	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

260 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58072);