#436820

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15573
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

574 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)(3,15,19,9)(4,20,25,41)(5,26)(6,13)(7,14)(8,12)(10,23,37,24)(11,27)(16,35)(17,46,49,55)(18,34,30,33)(21,42)(22,36,53,61)(28,39)(29,40)(31,44)(32,48,38,47)(43,57)(45,58)(50,54,56,52)(51,59)(60,63)(62,64);(1,3,6,18,8,19,7,30)(2,9,13,33,12,15,14,34)(4,21,23,16,25,44,24,45)(5,17,27,47,29,49,28,48)(10,35,20,31,37,58,41,42)(11,38,40,46,39,32,26,55)(22,51,52,43,53,63,54,64)(36,60,50,62,61,59,56,57);(1,4)(2,10,12,37)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31,15,42)(11,36,39,61)(13,20,14,41)(17,43)(18,44)(19,45)(21,30)(26,56,40,50)(27,52)(28,54)(29,53)(32,57,38,62)(33,58,34,35)(46,59,55,60)(47,63)(48,51)(49,64);(1,5,6,27,8,29,7,28)(2,11,13,40,12,39,14,26)(3,17,18,47,19,49,30,48)(4,22,23,52,25,53,24,54)(9,32,33,55,15,38,34,46)(10,36,20,50,37,61,41,56)(16,43,44,63,45,64,21,51)(31,57,58,60,42,62,35,59)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=e,d^2=f,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,g^b=gi,h^b=hi,g^c=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, 344016);