#150251

order	512
nilpotency class	5
isoclinism	isoclinic class of #150243
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

305 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 84 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)(9,10,12,18,16,23,34,22)(11,24,25,58,29,61,30,60)(13,35,31,67,39,53,70,71)(14,40,32,20,15,42,33,21)(17,44,45,26,49,62,48,64)(19,51,50,38,55,69,54,37)(27,65,41,59,28,66,43,52)(36,56,72,46,63,57,68,47);(1,3)(2,5)(4,8)(6,7)(9,11,16,29)(10,17,23,49)(12,30,34,25)(13,36,39,63)(14,41,15,43)(18,48,22,45)(19,52,55,59)(20,56,21,57)(24,35,61,53)(26,37,64,38)(27,33,28,32)(31,68,70,72)(40,46,42,47)(44,51,62,69)(50,66,54,65)(58,71,60,67);(9,12,16,34)(10,18,23,22)(11,25,29,30)(13,31,39,70)(14,32,15,33)(17,45,49,48)(19,50,55,54)(20,42,21,40)(24,58,61,60)(26,62,64,44)(27,41,28,43)(35,67,53,71)(36,72,63,68)(37,51,38,69)(46,57,47,56)(52,65,59,66)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57447);