#449974

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15954
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

989 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)(3,9,16,33)(4,19,24,48)(5,11)(6,13)(7,29)(8,12)(10,22,36,23)(14,34)(15,44,25,31)(17,32,18,40)(20,30)(21,50,53,47)(26,38)(27,37)(28,39)(35,52,62,54)(41,60)(42,61)(43,57)(45,59,46,58)(49,56)(51,63)(55,64);(1,3,6,18,8,16,7,17)(2,9,29,32,12,33,13,40)(4,20,22,42,24,43,23,14)(5,25,26,45,27,15,28,46)(10,34,48,57,36,61,19,30)(11,31,39,58,37,44,38,59)(21,51,52,55,53,49,54,41)(35,60,47,56,62,64,50,63);(1,4)(2,10,12,36)(3,14,16,42)(5,21)(6,22)(7,23)(8,24)(9,30)(11,35,37,62)(13,19,29,48)(15,41,25,55)(17,43,18,20)(26,52)(27,53)(28,54)(31,56)(32,34)(33,57)(38,50,39,47)(40,61)(44,63)(45,49,46,51)(58,64)(59,60);(1,5)(2,11)(3,15)(4,21)(6,28)(7,26)(8,27)(9,31)(10,35)(12,37)(13,39)(14,41)(16,25)(17,46)(18,45)(19,47)(20,49)(22,54)(23,52)(24,53)(29,38)(30,56)(32,59)(33,44)(34,60)(36,62)(40,58)(42,55)(43,51)(48,50)(57,63)(61,64)

Polycyclic presentation:

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

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, 357170);