#152855

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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	C₂³	#508
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

182 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 86 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)(5,6,10,16,12,18,32,17)(7,13,21,35,22,36,11,33)(8,23,26,51,28,53,57,52)(9,29,30,59,31,61,55,60)(14,37,38,56,40,65,39,25)(15,20,41,48,43,49,42,24)(19,44,46,66,47,67,27,58)(34,62,63,68,64,54,50,45);(2,4)(5,7,10,21,12,22,32,11)(6,13,16,35,18,36,17,33)(8,19,26,46,28,47,57,27)(9,20,30,48,31,49,55,24)(14,34,38,63,40,64,39,50)(15,29,41,59,43,61,42,60)(23,44,51,66,53,67,52,58)(25,45,37,62,56,68,65,54);(5,8,24,54,11,27,55,65,32,57,49,68,22,47,31,56,12,28,48,62,21,46,30,37,10,26,20,45,7,19,9,25)(6,14,29,44,13,34,41,51,16,38,59,66,35,63,43,53,18,40,61,67,36,64,42,52,17,39,60,58,33,50,15,23)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60051);