#152437

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₄×D₁₆
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

263 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 82 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)(5,6,11,18,13,21,44,20)(7,22,26,56,29,49,28,61)(8,30,27,66,34,53,33,67)(9,35,38,64,10,40,42,68)(12,45,23,19,46,57,37,48)(14,47,50,25,52,65,63,24)(15,41,54,31,51,36,62,32)(16,55,58,43,17,60,59,39);(2,4)(5,7,11,26,13,29,44,28)(6,14,18,50,21,52,20,63)(8,23,27,46,34,37,33,12)(9,36,38,32,10,41,42,31)(15,48,54,45,51,19,62,57)(16,30,58,66,17,53,59,67)(22,64,56,40,49,68,61,35)(24,55,47,43,25,60,65,39);(5,8)(6,15)(7,23)(9,24)(10,25)(11,27)(12,28)(13,34)(14,48)(16,56)(17,61)(18,54)(19,52)(20,62)(21,51)(22,59)(26,46)(29,37)(30,40)(31,39)(32,43)(33,44)(35,53)(36,55)(38,47)(41,60)(42,65)(45,50)(49,58)(57,63)(64,67)(66,68)

Polycyclic presentation:

2^9|a^2=f,b^2=f,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,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

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59633);