#153106

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×#2263	C₂
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

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

normal subgroups only

This group has 377 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)(7,19)(8,14)(9,16)(10,32)(11,35)(12,18)(13,21)(15,26)(17,27)(20,36)(22,50)(23,58)(24,51)(25,48)(28,40)(29,47)(30,42)(31,49)(33,43)(34,64)(37,55)(38,54)(39,56)(41,63)(44,62)(45,53)(46,52)(57,67)(59,65)(60,66)(61,68);(2,4)(5,7,12,24)(6,13,18,39)(8,25,30,61)(9,20,11,22)(10,33,34,57)(14,19,42,51)(15,43,45,67)(16,46,35,37)(17,48,49,68)(21,53,56,26)(23,31,59,27)(28,54,29,60)(32,38,64,66)(36,47,50,40)(41,58,44,65)(52,63,55,62);(6,14)(7,20)(8,26)(9,11)(10,27)(13,36)(15,17)(16,47)(18,42)(19,46)(21,54)(22,24)(23,59)(25,52)(28,63)(29,62)(30,53)(31,34)(32,41)(35,40)(37,51)(38,43)(39,50)(44,64)(45,49)(48,65)(55,61)(56,60)(58,68)(66,67)

Polycyclic presentation:

2^9|b^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, 60302);