#129265

order	512
nilpotency class	4
isoclinism	isoclinic class of #12707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1781
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8299
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,e^b=ei,f^b=fgi,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, 36461);