#355494

order	512
nilpotency class	3
isoclinism	isoclinic class of #21776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2308
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13784
prime radical	#3449	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1419 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)(4,9,20,30)(5,22,23,10)(6,12)(7,11)(14,34)(15,27)(17,36,42,28)(18,26)(19,29)(21,31,37,54)(24,48,49,32)(25,33)(35,40)(38,55,43,50)(39,56)(41,52)(44,51)(45,47)(46,53)(57,60)(58,62)(59,64,61,63);(1,3,6,17,7,16,25,42)(2,8,12,28,11,13,33,36)(4,18,21,44,20,14,37,39)(5,15,24,41,23,40,49,60)(9,26,31,51,30,34,54,56)(10,27,32,52,22,35,48,57)(19,43,46,61,45,38,58,59)(29,50,53,63,47,55,62,64);(1,4)(2,9,11,30)(3,14,16,18)(5,19)(6,21)(7,20)(8,26)(10,29,22,47)(12,31,33,54)(13,34)(15,38,40,43)(17,39,42,44)(23,45)(24,46)(25,37)(27,50)(28,51)(32,53,48,62)(35,55)(36,56)(41,59,60,61)(49,58)(52,63)(57,64);(1,5)(2,10)(3,15)(4,19)(6,24)(7,23)(8,27)(9,29)(11,22)(12,32)(13,35)(14,38)(16,40)(17,41)(18,43)(20,45)(21,46)(25,49)(26,50)(28,52)(30,47)(31,53)(33,48)(34,55)(36,57)(37,58)(39,59)(42,60)(44,61)(51,63)(54,62)(56,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317348, #72317353, #72317365
maximal normal subgroup of	#72317348, #72317353, #72317365

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262690);