#130789

order	512
nilpotency class	4
isoclinism	isoclinic class of #130786
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8375
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#61090	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

807 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 807 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=ei,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, #72250052, #72250217, …
maximal normal subgroup of	#72250043, #72250052, #72250217, …
maximal quotient of	#72250432

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37985);