#129229

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	#8298
prime radical	C₂×#585	C₄
LCS	G ▷ C₂²×C₄ ▷ 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,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=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, 36425);