#152278

order	512
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×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

276 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 98 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)(7,17)(8,23)(9,28)(10,31)(11,35)(12,18)(13,20)(14,25)(15,30)(16,33)(19,48)(21,52)(22,47)(24,57)(26,56)(27,58)(29,38)(32,46)(34,64)(36,44)(37,61)(39,59)(40,65)(41,53)(42,60)(43,54)(45,51)(49,55)(50,68)(62,63)(66,67);(5,7,9,20,12,22,30,11)(6,13,15,17,18,35,28,47)(8,19,24,49,27,50,26,36)(10,21,29,51,34,54,62,33)(14,37,39,46,42,40,41,67)(16,38,43,31,45,63,52,64)(23,55,56,48,58,44,57,68)(25,32,53,61,60,66,59,65);(5,8)(6,14)(7,19)(9,24)(10,32)(11,36)(12,27)(13,37)(15,39)(16,44)(17,46)(18,42)(20,49)(21,53)(22,50)(23,31)(25,33)(26,30)(28,41)(29,61)(34,66)(35,40)(38,57)(43,68)(45,55)(47,67)(48,52)(51,60)(54,59)(56,63)(58,64)(62,65)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59474);