#512719

order	512
nilpotency class	5
isoclinism	isoclinic class of #33982
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#41	D₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

420 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 274 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 — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,e^2=gh,g^2=hi,h^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419915);