#130779

order	512
nilpotency class	4
isoclinism	isoclinic class of #130776
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	#61051	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

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

normal subgroups only

This group has 811 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,f^2=h,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=fg,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250527, #72250531, #72250556, …
maximal normal subgroup of	#72250527, #72250531, #72250556, …
maximal quotient of	#72250865

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37975);