#129266

order	512
nilpotency class	4
isoclinism	isoclinic class of #12705
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1781
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8299
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

799 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 799 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=ei,f^b=fg,d^c=di,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36462);