#355934

order	512
nilpotency class	3
isoclinism	isoclinic class of #21776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2308
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13794
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2329 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)(3,13,16,8)(4,9)(5,21,22,10)(6,12)(7,11)(14,25,37,28)(15,26)(17,35,41,27)(18,42,43,29)(19,31)(20,30)(23,46,47,32)(24,33)(34,39)(36,51)(38,49,58,52)(40,50)(44,60,61,53)(45,54)(48,56)(55,59)(57,63)(62,64);(1,3)(2,8)(4,14)(5,15)(6,17)(7,16)(9,28)(10,26)(11,13)(12,27)(18,36)(19,38)(20,37)(21,34)(22,39)(23,40)(24,41)(25,30)(29,51)(31,52)(32,50)(33,35)(42,48)(43,56)(44,57)(45,58)(46,55)(47,59)(49,54)(53,63)(60,62)(61,64);(1,4,6,19,7,20,24,45)(2,9,12,31,11,30,33,54)(3,14,17,38,16,37,41,58)(5,18,23,44,22,43,47,61)(8,25,27,49,13,28,35,52)(10,29,32,53,21,42,46,60)(15,36,40,57,39,56,59,64)(26,48,50,62,34,51,55,63);(1,5)(2,10)(3,15)(4,18)(6,23)(7,22)(8,26)(9,29)(11,21)(12,32)(13,34)(14,36)(16,39)(17,40)(19,44)(20,43)(24,47)(25,48)(27,50)(28,51)(30,42)(31,53)(33,46)(35,55)(37,56)(38,57)(41,59)(45,61)(49,62)(52,63)(54,60)(58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317343, #72317361, #72317365
maximal normal subgroup of	#72317343, #72317361, #72317365

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263130);