#130293

order	512
nilpotency class	4
isoclinism	isoclinic class of #130292
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8350
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#62284	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 590 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240824, #72250762, #72250804, …
maximal normal subgroup of	#72240824, #72250762, #72250804, …
maximal quotient of	#72240706

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37489);