#513170

order	512
nilpotency class	5
isoclinism	isoclinic class of D₄×D₃₂
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
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)(6,12)(7,17)(8,23)(9,26)(10,30)(13,18)(14,42)(15,46)(16,37)(20,21)(22,50)(24,29)(25,48)(27,61)(28,34)(31,43)(32,36)(33,49)(35,44)(39,55)(40,56)(41,53)(45,62)(47,52)(51,64)(54,59)(57,58)(60,63)(65,68)(66,67);(5,6,11,16)(7,18,22,53)(8,24,28,51)(9,29,31,64)(10,32,30,15)(12,34,37,23)(13,38,41,19)(14,43,45,26)(17,35,50,65)(20,55,49,66)(21,56,33,52)(25,44,60,68)(27,62,57,42)(36,58,46,61)(39,63,67,48)(40,54,47,59);(2,4);(5,7,11,22)(6,13,16,41)(8,25,28,60)(9,17,31,50)(10,33,30,21)(12,35,37,65)(14,44,45,68)(15,47,32,40)(18,51,53,24)(19,34,38,23)(20,43,49,26)(27,63,57,48)(29,39,64,67)(36,66,46,55)(42,56,62,52)(54,58,59,61)

Polycyclic presentation:

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

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420366);