#153294

order	512
nilpotency class	5
isoclinism	isoclinic class of #153293
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13422
commutator, Frattini, cosocle	#532	C₂³
prime radical	#3018	C₂²
LCS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

387 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fgh,g^a=gi,c^b=cf,d^b=di,e^b=eg,g^b=gi,h^b=hi,e^c=ehi,f^c=fi,g^c=gh,f^e=fh,g^f=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60490);