#452885

order	512
nilpotency class	3
isoclinism	isoclinic class of #424595
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16069
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

577 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)(3,15,20,9)(4,21,25,38)(5,26)(6,13)(7,14)(8,12)(10,24,37,30)(11,27)(16,43)(17,47,50,55)(18,34,19,33)(22,31)(23,36,54,61)(28,40)(29,41)(32,49,39,48)(35,46)(42,45)(44,58)(51,57,56,53)(52,59)(60,63)(62,64);(1,3,7,19,8,20,6,18)(2,9,14,34,12,15,13,33)(4,22,24,46,25,45,30,16)(5,17,28,49,29,50,27,48)(10,35,21,42,37,43,38,31)(11,39,26,55,40,32,41,47)(23,52,53,64,54,63,57,44)(36,60,51,58,61,59,56,62);(1,4)(2,10,12,37)(3,16)(5,23)(6,24)(7,30)(8,25)(9,31,15,42)(11,36,40,61)(13,21,14,38)(17,44)(18,22)(19,45)(20,46)(26,56,41,51)(27,53)(28,57)(29,54)(32,58,39,62)(33,35,34,43)(47,60,55,59)(48,52)(49,63)(50,64);(1,5,6,27,8,29,7,28)(2,11,14,26,12,40,13,41)(3,17,18,48,20,50,19,49)(4,23,24,53,25,54,30,57)(9,32,34,47,15,39,33,55)(10,36,38,56,37,61,21,51)(16,44,22,52,46,64,45,63)(31,58,43,60,42,62,35,59)

Polycyclic presentation:

2^9|a^2=egh,b^2=g,c^2=e,d^2=ef,f^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,h^b=hi,g^c=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, 360081);