#153270

order	512
nilpotency class	5
isoclinism	isoclinic class of #153105
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 300 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,13,20)(7,11,24,40)(8,25,30,60)(9,19,12,16)(10,33,38,55)(14,18,32,22)(15,43,47,28)(17,49,34,36)(21,53,46,65)(23,31,57,37)(26,39,59,67)(27,56,41,58)(29,62,64,61)(35,54,44,51)(42,48,66,50)(45,63,52,68);(5,7,13,24)(6,14,20,32)(8,26,30,59)(9,18,12,22)(10,34,38,17)(11,16,40,19)(15,44,47,35)(21,43,46,28)(23,49,57,36)(25,29,60,64)(27,62,41,61)(31,48,37,50)(33,42,55,66)(39,58,67,56)(45,65,52,53)(51,68,54,63);(2,4)(5,8,9,27,13,30,12,41)(6,15,16,45,20,47,19,52)(7,21,22,54,24,46,18,51)(10,35,31,65,38,44,37,53)(11,39,32,64,40,67,14,29)(17,26,48,61,34,59,50,62)(23,56,55,25,57,58,33,60)(28,42,63,36,43,66,68,49)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=d,d^2=i,e^2=gh,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fg,g^a=gh,h^a=hi,c^b=cf,d^b=di,e^b=eg,f^b=fg,g^b=gh,h^b=hi,e^c=egi,f^c=fgi,g^c=gh,h^c=hi

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, 60466);