#424610

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15323
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1203 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)(5,22)(6,25)(7,11)(10,24)(12,28)(14,29,18,32)(15,42,43,54)(17,31,26,38)(19,48)(20,34)(21,52)(23,39)(27,37)(30,44,36,45)(33,50)(35,51)(40,61,46,64)(41,58,47,56)(49,60)(53,59)(55,63,57,62);(1,3)(2,8)(4,18)(5,15)(6,26)(7,16)(9,32)(10,36)(11,13)(12,38)(14,20)(17,28)(19,40)(21,41)(22,54)(23,43)(24,45)(25,31)(27,44)(29,34)(30,37)(33,57)(35,56)(39,42)(46,49)(47,51)(48,61)(50,63)(52,58)(53,62)(55,59)(60,64);(1,4)(2,9)(3,14,16,18)(5,19)(6,21)(7,20)(8,29,13,32)(10,33)(11,34)(12,35)(15,40,43,46)(17,41,26,47)(22,48)(23,49)(24,50)(25,52)(27,53)(28,51)(30,55,36,57)(31,56,38,58)(37,59)(39,60)(42,64,54,61)(44,62,45,63);(1,5)(2,10)(3,15)(4,19)(6,27)(7,23)(8,30)(9,33)(11,37)(12,39)(13,36)(14,40)(16,43)(17,45)(18,46)(20,49)(21,53)(22,25)(24,28)(26,44)(29,55)(31,42)(32,57)(34,59)(35,60)(38,54)(41,63)(47,62)(48,52)(50,51)(56,64)(58,61)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331806);