#419467

order	512
nilpotency class	3
isoclinism	isoclinic class of #417669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15270
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1299 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)(3,14,17,9)(4,10,22,35)(5,25,26,39)(6,28)(7,13)(8,12)(11,27,38,29)(15,30)(16,46)(18,32,19,40)(20,50)(21,52)(23,41,24,36)(31,47)(33,44)(34,55)(37,49)(42,45)(43,60,63,59)(48,56)(51,62,64,58)(53,61)(54,57);(1,3)(2,9)(4,20)(5,16)(6,19)(7,18)(8,17)(10,33)(11,37)(12,14)(13,40)(15,23)(21,51)(22,44)(24,45)(25,56)(26,48)(27,49)(28,32)(29,47)(30,36)(31,38)(34,60)(35,50)(39,46)(41,42)(43,53)(52,58)(54,64)(55,63)(57,62)(59,61);(1,4)(2,10,12,35)(3,15)(5,21)(6,24)(7,23)(8,22)(9,30,14,42)(11,34,38,61)(13,41,28,36)(16,43)(17,45)(18,44)(19,20)(25,57,39,52)(26,54)(27,55)(29,53)(31,58,37,62)(32,50,40,33)(46,60,56,59)(47,64)(48,63)(49,51);(1,5)(2,11)(3,16)(4,21)(6,27)(7,29)(8,26)(9,31)(10,34)(12,38)(13,39)(14,37)(15,43)(17,48)(18,47)(19,49)(20,51)(22,54)(23,53)(24,55)(25,28)(30,58)(32,46)(33,59)(35,61)(36,57)(40,56)(41,52)(42,62)(44,64)(45,63)(50,60)

Polycyclic presentation:

2^9|a^2=ef,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 326663);