#153269

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	C₂²×D₁₆	C₂²
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

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

normal subgroups only

This group has 310 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)(5,6,12,18)(7,19,23,53)(8,13,11,15)(9,26,31,64)(10,32,34,44)(14,38,42,20)(16,45,48,27)(17,25,49,29)(21,39,35,51)(22,58,60,67)(24,46,30,63)(28,65,56,50)(33,52,66,47)(36,57,40,55)(37,54,43,62)(41,61,68,59);(6,13)(7,20)(8,11)(9,27)(10,29)(14,39)(15,18)(16,46)(17,49)(19,36)(21,57)(22,59)(23,38)(24,54)(25,34)(26,37)(28,66)(30,62)(31,45)(33,56)(35,55)(40,53)(41,52)(42,51)(43,64)(47,68)(48,63)(50,67)(58,65)(60,61);(2,4)(5,7,8,21,12,23,11,35)(6,14,15,40,18,42,13,36)(9,28,24,61,31,56,30,59)(10,33,25,60,34,66,29,22)(16,47,43,67,48,52,37,58)(17,50,44,68,49,65,32,41)(19,46,51,26,53,63,39,64)(20,54,55,27,38,62,57,45)

Polycyclic presentation:

2^9|a^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, 60465);