#513179

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
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

436 conjugacy classes of subgroups.

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

Relationships

maximal subgroup of	#72234341
maximal normal subgroup of	#72234341

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420375);