#383754

order	512
nilpotency class	3
isoclinism	isoclinic class of #383753
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14363
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1303 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264934, #72264941, #72264989
maximal normal subgroup of	#72264934, #72264941, #72264989
maximal quotient of	#72314029

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290950);