#432667

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15463
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1090 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,g^c=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, 339863);