#129228

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	#8298
prime radical	C₂×#585	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1351 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 1351 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:

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

Polycyclic presentation:

2^9|a^2=f,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,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, 36424);