#317323

order	512
nilpotency class	3
isoclinism	isoclinic class of #317315
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#553
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

439 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,f^b=fh,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224519);