#513168

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

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

normal subgroups only

This group has 434 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)(5,6,13,8)(7,19,24,54)(9,16,31,26)(10,15,35,25)(11,36,39,61)(12,28,38,18)(14,42,20,21)(17,49,27,33)(22,57,58,68)(23,59,40,53)(29,46,34,52)(30,45,41,48)(32,50,65,62)(37,47,67,60)(43,66,55,63)(44,56,51,64);(5,7)(6,14)(8,20)(9,21)(10,32)(11,37)(12,40)(13,24)(15,19)(16,47)(17,50)(18,51)(22,33)(23,38)(25,54)(26,60)(27,62)(28,44)(29,63)(30,64)(31,42)(34,66)(35,65)(36,43)(39,67)(41,56)(45,57)(46,59)(48,68)(49,58)(52,53)(55,61);(2,4)(5,8,13,6)(7,20,24,14)(9,25,31,15)(10,26,35,16)(11,27,39,17)(12,28,38,18)(19,21,54,42)(22,55,58,43)(23,51,40,44)(29,48,34,45)(30,52,41,46)(32,60,65,47)(33,61,49,36)(37,62,67,50)(53,56,59,64)(57,63,68,66);(5,9)(6,15)(8,25)(10,11)(12,41)(13,31)(16,17)(18,52)(19,47)(21,32)(22,37)(23,40)(26,27)(28,46)(29,33)(30,38)(34,49)(35,39)(36,45)(42,65)(43,50)(44,51)(48,61)(53,68)(54,60)(55,62)(56,66)(57,59)(58,67)(63,64)

Polycyclic presentation:

2^9|a^2=i,c^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, 420364);