#130289

order	512
nilpotency class	4
isoclinism	isoclinic class of #130284
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	#62274	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

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

normal subgroups only

This group has 599 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250039, #72250070, #72250072, …
maximal normal subgroup of	#72250039, #72250070, #72250072, …
maximal quotient of	#72250401, #72250687, #72280439

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37485);