#151183

order	512
nilpotency class	5
isoclinism	isoclinic class of #151182
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12102
commutator, Frattini, cosocle	C₂×M₃₂	C₂³
prime radical	#13309	C₂
LCS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2112 ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

291 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=fh,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gi,h^a=hi,d^b=df,e^b=egi,f^b=fh,h^b=hi,d^c=dg,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320999, #72321024, #72321032
maximal normal subgroup of	#72320999, #72321024, #72321032

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58379);