#513174

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

442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,11,16)(7,17,18,48)(8,23,27,59)(9,28,31,47)(10,32,30,15)(12,34,35,20)(13,38,40,25)(14,41,45,24)(19,39,50,66)(21,55,51,68)(22,56,33,49)(26,62,60,58)(29,42,64,67)(36,57,65,53)(37,54,46,52)(43,63,44,61);(5,7)(6,12)(8,19)(9,29)(10,33)(11,18)(13,39)(14,42)(15,46)(16,35)(17,23)(20,24)(21,25)(22,30)(26,57)(27,50)(28,36)(31,64)(32,37)(34,41)(38,51)(40,66)(43,55)(44,68)(45,67)(47,65)(48,59)(49,62)(52,61)(53,60)(54,63)(56,58);(2,4)(7,18)(12,35)(17,48)(19,50)(20,34)(21,51)(22,33)(29,64)(36,65)(37,46)(39,66)(42,67)(49,56)(52,54)(53,57)(55,68);(6,13)(7,19)(8,24)(9,25)(10,30)(12,17)(14,43)(15,47)(16,40)(18,50)(20,21)(22,57)(26,63)(27,41)(28,32)(29,52)(31,38)(33,53)(34,51)(35,48)(36,55)(37,56)(39,42)(44,45)(46,49)(54,64)(58,62)(60,61)(65,68)(66,67)

Polycyclic presentation:

2^9|a^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	#72234374, #72322380, #72322420
maximal normal subgroup of	#72234374, #72322380, #72322420

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420370);