#153588

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

122 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 256

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60784);