#138942

order	512
nilpotency class	4
isoclinism	isoclinic class of #138936
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9259
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	C₂×#576	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

520 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=gh,e^2=ghi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=egh,f^c=fgh,h^c=hi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280935
maximal normal subgroup of	#72280935

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46138);