#130846

order	512
nilpotency class	4
isoclinism	isoclinic class of #130844
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8376
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#61290	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

554 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fgh,h^b=hi,f^c=fgi,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250043, #72250061, #72250202, …
maximal normal subgroup of	#72250043, #72250061, #72250202, …
maximal quotient of	#72250426

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38042);