#152847

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	D₄×C₁₆
prime radical	C₂³	#504
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

194 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 110 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)(5,6,10,17,11,18,32,44,12,16,33,42,29,43,56,64,13,19,34,45,27,15,40,41,9,30,31,57,8,26,28,55)(7,14,22,37,23,38,52,62,24,36,53,60,49,61,66,68,25,39,54,63,47,35,58,59,21,50,51,67,20,46,48,65);(2,4)(5,7,10,22,11,23,32,52,12,24,33,53,29,49,56,66,13,25,34,54,27,47,40,58,9,21,31,51,8,20,28,48)(6,14,17,37,18,38,44,62,16,36,42,60,43,61,64,68,19,39,45,63,15,35,41,59,30,50,57,67,26,46,55,65);(5,8,9,27,13,29,12,11)(6,15,16,26,19,18,30,43)(7,20,21,47,25,49,24,23)(10,28,31,40,34,56,33,32)(14,35,36,46,39,38,50,61)(17,41,42,55,45,44,57,64)(22,48,51,58,54,66,53,52)(37,59,60,65,63,62,67,68)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=e,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60043);