#152997

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

252 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,16,21,17,22,27,34,18,23,28,35,29,36,40,43,19,15,20,24,30,25,32,37,31,26,33,38,41,39,42,44);(1,2,3,4)(6,9)(10,11)(13,15,16,24,17,25,27,37,18,26,28,38,29,39,40,44,19,14,20,21,30,22,32,34,31,23,33,35,41,36,42,43);(5,7,8,12)(6,10,9,11)

Polycyclic presentation:

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

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, 60193);