#427435

order	512
nilpotency class	3
isoclinism	isoclinic class of #427434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15369
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1198 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334631);