#153107

order	512
nilpotency class	5
isoclinism	isoclinic class of #153105
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	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

367 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 76 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)(7,20)(8,27)(9,31)(10,17)(11,39)(12,41)(13,19)(14,22)(15,28)(16,35)(18,21)(23,47)(24,60)(25,61)(26,55)(29,62)(30,46)(32,51)(33,44)(34,65)(36,67)(37,53)(38,52)(40,68)(42,58)(43,64)(45,63)(48,66)(49,57)(50,59)(54,56);(2,4)(5,7,13,26)(6,14,19,44)(8,11,30,40)(9,32,37,63)(10,25,12,23)(15,39,43,68)(16,46,49,27)(17,50,41,42)(18,51,52,45)(20,53,55,31)(21,24,38,29)(22,57,33,35)(28,34,64,36)(47,65,61,67)(48,62,54,60)(56,58,66,59);(5,8,13,30)(6,15,19,43)(7,21,26,38)(9,33,37,22)(10,24,12,29)(11,23,40,25)(14,18,44,52)(16,47,49,61)(17,51,41,45)(20,54,55,48)(27,53,46,31)(28,57,64,35)(32,56,63,66)(34,58,36,59)(39,42,68,50)(60,67,62,65)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gh,e^2=gh,f^2=i,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,f^b=fi,g^b=gh,h^b=hi,d^c=dg,e^c=eg,f^c=fi,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, 60303);