#152878

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²×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	C₂×C₂²⋊C₁₆
prime radical	#532	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 290 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,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, 60074);