#423182

order	512
nilpotency class	3
isoclinism	isoclinic class of #420416
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	#60422	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1292 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)(4,10)(5,11)(6,29)(7,13)(8,12)(9,17)(15,33)(16,44)(18,32)(19,40)(20,50)(21,52)(22,41)(23,35)(24,36)(25,57)(26,42)(27,38)(28,39)(30,46)(31,49)(34,55)(37,48)(43,47)(45,63)(51,59)(53,60)(54,61)(56,62)(58,64);(1,3)(2,9)(4,20)(5,25)(6,18)(7,19)(8,17)(10,33)(11,37)(12,40)(13,32)(14,29)(15,23)(16,26)(21,53)(22,47)(24,46)(27,48)(28,49)(30,35)(31,38)(34,59)(36,43)(39,44)(41,50)(42,57)(45,55)(51,54)(52,64)(56,58)(60,61)(62,63);(1,4,8,22,7,24,6,23)(2,10,29,35,13,36,12,41)(3,15,17,20,19,47,18,46)(5,21,26,56,28,55,27,54)(9,30,14,43,32,50,40,33)(11,34,38,62,39,52,42,61)(16,45,49,51,48,53,25,58)(31,59,44,63,57,64,37,60);(1,5)(2,11)(3,16)(4,21)(6,27)(7,28)(8,26)(9,31)(10,34)(12,42)(13,39)(14,44)(15,45)(17,49)(18,25)(19,48)(20,51)(22,56)(23,54)(24,55)(29,38)(30,59)(32,57)(33,60)(35,62)(36,52)(37,40)(41,61)(43,63)(46,58)(47,53)(50,64)

Polycyclic presentation:

2^9|c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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

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, 330378);