#146498

order	512
nilpotency class	4
isoclinism	isoclinic class of #12725
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂⁴	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#10726
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

817 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 817 conjugacy classes of subgroups — too many to draw. Only its 184 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)(3,13)(4,18)(5,22,23,10)(6,11)(7,12)(8,15,27,40)(9,20)(14,36)(16,34)(17,35)(19,29,46,44)(21,45)(24,49,50,31)(25,32)(26,51,43,37)(28,41,53,61)(30,48)(33,38)(39,58)(42,56)(47,54,63,57)(52,64,62,59)(55,60);(1,3,6,16,7,17,25,42)(2,8,11,28,12,27,32,53)(4,14,21,39,20,38,48,60)(5,15,24,41,23,40,50,61)(9,26,30,52,18,43,45,62)(10,13,31,34,22,35,49,56)(19,37,47,59,46,51,63,64)(29,33,54,55,44,36,57,58);(1,4)(2,9,12,18)(3,14)(5,19)(6,21)(7,20)(8,26,27,43)(10,29,22,44)(11,30,32,45)(13,33,35,36)(15,37)(16,39)(17,38)(23,46)(24,47)(25,48)(28,52,53,62)(31,54,49,57)(34,55,56,58)(40,51)(41,59)(42,60)(50,63)(61,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,f^b=fh,e^c=ei,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, 53694);