#153362

order	512
nilpotency class	5
isoclinism	isoclinic class of #153255
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2233
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

203 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 68 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)(5,6,10,12,13,18,9,16)(7,19,22,14,25,42,21,24)(8,26,28,30,31,40,33,15)(11,35,34,45,39,17,32,38)(20,50,52,68,53,41,55,48)(23,56,46,36,58,49,54,43)(27,44,60,61,62,67,63,57)(29,65,64,37,51,59,47,66);(5,7,13,25)(6,14,18,24)(8,27,31,62)(9,22,10,21)(11,36,39,43)(12,19,16,42)(15,44,30,67)(17,46,35,54)(20,51,53,29)(23,45,58,38)(26,57,40,61)(28,63,33,60)(32,49,34,56)(37,41,66,50)(47,52,64,55)(48,59,68,65);(2,4)(5,8,9,28,13,31,10,33)(6,15,16,26,18,30,12,40)(7,20,21,52,25,53,22,55)(11,37,32,59,39,66,34,65)(14,41,19,48,24,50,42,68)(17,47,45,29,35,64,38,51)(23,57,54,44,58,61,46,67)(27,49,60,36,62,56,63,43)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60558);