#146185

order	512
nilpotency class	4
isoclinism	isoclinic class of #144056
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1697
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10658
prime radical	C₂³	#473
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

179 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 179 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=d,c^2=dg,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,c^b=cf,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=eg,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53381);