#417980

order	512
nilpotency class	3
isoclinism	isoclinic class of #417980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15259
prime radical	#60398	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1936 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)(3,9)(4,19,22,36)(5,24)(6,12)(7,28)(8,13)(10,23,35,27)(11,25)(14,39,20,49)(15,42)(16,32)(17,33)(18,46)(21,51,52,34)(26,37)(29,38)(30,41,56,45)(31,43)(40,59,50,55)(44,57)(47,58)(48,53,60,54)(61,62,64,63);(1,3)(2,9)(4,20,22,14)(5,15)(6,16)(7,18)(8,17)(10,30,35,56)(11,31)(12,32)(13,33)(19,39,36,49)(21,40,52,50)(23,45,27,41)(24,42)(25,43)(26,44)(28,46)(29,47)(34,59,51,55)(37,57)(38,58)(48,64,60,61)(53,62,54,63);(1,4)(2,10)(3,14)(5,21)(6,27)(7,23)(8,22)(9,30)(11,34)(12,36)(13,35)(15,40)(16,45)(17,20)(18,41)(19,28)(24,48)(25,54)(26,52)(29,53)(31,55)(32,49)(33,56)(37,60)(38,51)(39,46)(42,61)(43,63)(44,50)(47,62)(57,64)(58,59);(1,5)(2,11)(3,15)(4,21)(6,25)(7,29)(8,26)(9,31)(10,34)(12,37)(13,38)(14,40)(16,43)(17,44)(18,47)(19,48)(20,50)(22,52)(23,53)(24,28)(27,54)(30,55)(32,57)(33,58)(35,51)(36,60)(39,61)(41,62)(42,46)(45,63)(49,64)(56,59)

Polycyclic presentation:

2^9|a^2=e,b^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=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, 325176);