#153104

order	512
nilpotency class	5
isoclinism	isoclinic class of #153078
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

373 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 72
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,4)(9,10,16,22)(11,23,12,27)(13,31,37,57)(14,38,42,71)(15,43,41,21)(17,28,18,24)(19,48,52,35)(20,53,56,34)(25,60,29,63)(26,61,30,58)(32,54,33,55)(36,70,69,65)(39,49,40,50)(44,67,46,66)(45,59,47,62)(51,68,64,72);(2,4)(5,6)(7,8)(9,11,16,12)(10,17,22,18)(13,32,37,33)(14,39,42,40)(15,30,41,26)(19,49,52,50)(20,54,56,55)(21,47,43,45)(23,57,27,31)(24,48,28,35)(25,53,29,34)(36,67,69,66)(38,44,71,46)(51,63,64,60)(58,65,61,70)(59,68,62,72);(6,8)(9,12,16,11)(10,18,22,17)(13,33,37,32)(14,40,42,39)(15,26,41,30)(19,50,52,49)(20,55,56,54)(21,45,43,47)(23,31,27,57)(24,35,28,48)(25,34,29,53)(36,66,69,67)(38,46,71,44)(51,60,64,63)(58,70,61,65)(59,72,62,68)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=gh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gh,h^b=hi,d^c=dg,e^c=eg,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60300);