#153347

order	512
nilpotency class	5
isoclinism	isoclinic class of #153083
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#508
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 202 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=de,b^2=dg,c^2=i,d^2=gh,e^2=gi,f^2=g,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,h^a=hi,c^b=cf,d^c=dg,e^c=eghi,f^c=fgh,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, 60543);