#513172

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

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

normal subgroups only

This group has 444 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)(7,18)(8,22)(9,28)(11,21)(13,38)(14,23)(15,37)(16,45)(17,35)(19,50)(20,54)(25,53)(26,27)(29,33)(30,58)(31,34)(32,46)(36,47)(39,63)(40,59)(42,60)(43,62)(44,55)(48,51)(49,57)(52,61)(56,64)(65,68)(66,67);(5,6,10,15)(7,13,18,17)(8,23,25,60)(9,29,31,59)(11,33,32,40)(12,34,37,28)(14,41,42,24)(16,46,44,21)(19,51,27,63)(20,55,49,45)(22,36,53,65)(26,62,50,66)(30,47,64,68)(35,54,38,57)(39,61,48,52)(43,56,67,58);(2,4)(5,7,10,18)(6,13,15,17)(8,19,25,27)(9,20,31,49)(11,21,32,46)(12,35,37,38)(14,39,42,48)(16,40,44,33)(22,26,53,50)(23,51,60,63)(24,52,41,61)(28,57,34,54)(29,55,59,45)(30,56,64,58)(36,62,65,66)(43,47,67,68);(5,8,10,25)(6,14,15,42)(7,19,18,27)(9,30,31,64)(11,22,32,53)(12,36,37,65)(13,39,17,48)(16,47,44,68)(20,56,49,58)(21,26,46,50)(23,59,60,29)(24,34,41,28)(33,43,40,67)(35,62,38,66)(45,63,55,51)(52,54,61,57)

Polycyclic presentation:

2^9|b^2=i,c^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.

Relationships

maximal subgroup of	#72234341
maximal normal subgroup of	#72234341

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420368);