#152810

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂×C₄

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	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

217 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 94 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)(5,6,10,17,11,15,32,40,12,16,33,36,8,27,28,55,13,18,34,42,29,41,57,61,9,31,20,46,30,56,47,65)(7,19,23,44,24,43,52,62,25,39,53,60,21,48,49,66,26,45,54,64,50,63,67,68,22,14,35,38,51,37,58,59);(5,7,20,35,29,50,28,49,12,25,10,23,30,51,57,67,13,26,33,53,11,24,47,58,9,22,34,54,8,21,32,52)(6,14,36,44,41,48,65,68,16,19,42,60,56,63,40,59,18,39,46,64,15,37,55,62,31,45,17,38,27,43,61,66);(2,4)(5,8,9,11,13,30,12,29)(6,15,16,27,18,41,31,56)(7,21,22,24,26,51,25,50)(10,28,20,32,34,47,33,57)(14,37,19,43,39,48,45,63)(17,40,36,55,42,61,46,65)(23,49,35,52,54,58,53,67)(38,59,44,62,60,66,64,68)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=d,d^2=i,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,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, 60006);