#152203

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

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	C₂×C₄⋊C₁₆
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

310 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 130 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)(5,6,9,15,10,16,27,38,11,14,28,36,30,37,53,59,12,17,29,39,31,40,54,60,8,24,25,50,26,51,52,66)(7,18,21,42,22,43,47,63,23,33,48,56,49,57,65,67,19,41,44,61,45,62,64,68,20,13,32,34,46,35,55,58);(5,7,8,20,12,19,11,23)(6,13,14,18,17,33,24,41)(9,21,25,32,29,44,28,48)(10,22,26,46,31,45,30,49)(15,34,36,42,39,56,50,61)(16,35,37,43,40,57,51,62)(27,47,52,55,54,64,53,65)(38,58,59,63,60,67,66,68);(2,4)(7,19)(13,33)(18,41)(20,23)(21,44)(22,45)(32,48)(34,56)(35,57)(42,61)(43,62)(46,49)(47,64)(55,65)(58,67)(63,68)

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59399);