#513187

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33853	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 122 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)(5,6,12,18)(7,19,20,56)(8,25,30,35)(9,29,36,43)(10,37,41,32)(11,27,40,17)(13,44,45,22)(14,39,49,51)(15,48,28,34)(16,52,54,33)(21,59,55,47)(23,60,58,67)(24,62,42,26)(31,53,64,68)(38,50,66,65)(46,57,61,63);(5,7,12,20)(6,13,18,45)(8,26,30,62)(9,31,36,64)(10,38,41,66)(11,42,40,24)(14,47,49,59)(15,50,28,65)(16,53,54,68)(17,55,27,21)(19,43,56,29)(22,48,44,34)(23,52,58,33)(25,63,35,57)(32,61,37,46)(39,60,51,67);(2,4)(7,20)(13,45)(19,56)(21,55)(22,44)(23,58)(24,42)(26,62)(31,64)(38,66)(46,61)(47,59)(50,65)(53,68)(57,63)(60,67);(5,8,12,30)(6,14,18,49)(7,21,20,55)(9,32,36,37)(10,39,41,51)(11,43,40,29)(13,42,45,24)(15,33,28,52)(16,35,54,25)(17,34,27,48)(19,57,56,63)(22,60,44,67)(23,61,58,46)(26,64,62,31)(38,53,66,68)(47,65,59,50)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=de,e^b=eg,g^b=gh,h^b=hi,e^d=eg,g^d=gh,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420383);