#152820

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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

212 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=d,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,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, 60016);