#423183

order	512
nilpotency class	3
isoclinism	isoclinic class of #420517
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15312
prime radical	#60469	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1276 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)(3,15)(4,10)(5,25)(6,14)(7,13)(8,12)(9,18)(11,28)(16,44)(17,32)(19,33)(20,31)(21,51)(22,38)(23,37)(24,39)(26,43)(27,42)(29,41)(30,34)(35,46)(36,54)(40,49)(45,57)(47,55)(48,58)(50,59)(52,61)(53,56)(60,64)(62,63);(1,3)(2,9)(4,20)(5,26)(6,30)(7,19)(8,18)(10,35)(11,40)(12,33)(13,34)(14,15)(16,22)(17,27)(21,45)(23,46)(24,48)(25,55)(28,47)(29,49)(31,38)(32,41)(36,60)(37,58)(39,44)(42,43)(50,52)(51,62)(53,64)(54,63)(56,59)(57,61);(1,4,8,22,7,24,6,23)(2,10,14,37,13,39,12,38)(3,16,18,48,19,46,30,20)(5,21,27,54,29,53,28,52)(9,31,15,35,34,58,33,44)(11,36,41,51,42,61,25,56)(17,45,49,63,47,64,26,50)(32,57,43,59,55,60,40,62);(1,5)(2,11)(3,17)(4,21)(6,28)(7,29)(8,27)(9,32)(10,36)(12,25)(13,42)(14,41)(15,43)(16,45)(18,49)(19,47)(20,50)(22,54)(23,52)(24,53)(26,30)(31,57)(33,40)(34,55)(35,59)(37,51)(38,56)(39,61)(44,62)(46,64)(48,63)(58,60)

Polycyclic presentation:

2^9|c^2=ef,e^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,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 330379);