#474239

order	512
nilpotency class	3
isoclinism	isoclinic class of #455699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2797
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16918
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

532 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)(3,14,18,41)(4,9)(5,25)(6,11)(7,13)(8,17,33,19)(10,27)(12,30)(15,43,20,52)(16,32,49,61)(21,38)(22,36)(23,42)(24,37)(26,56)(28,39)(29,40)(31,46,34,51)(35,55)(44,48,60,50)(45,62,53,58)(47,64,57,59)(54,63);(1,3,7,18)(2,8)(4,20)(5,16,28,49)(6,17,30,19)(9,34,37,31)(10,32)(11,14)(12,41)(13,33)(15,24)(21,53)(22,51)(23,46)(25,44)(26,57)(27,48,29,50)(35,62,56,58)(36,52,42,43)(38,64,63,59)(39,60)(40,61)(45,54)(47,55);(1,4,6,22,7,24,30,23)(2,9,11,36,13,37,12,42)(3,15,17,46,18,20,19,51)(5,26,27,54,28,55,29,21)(8,31,14,43,33,34,41,52)(10,38,39,56,40,63,25,35)(16,47,48,53,49,57,50,45)(32,59,60,62,61,64,44,58);(1,5,6,29,7,28,30,27)(2,10,11,25,13,40,12,39)(3,16,17,50,18,49,19,48)(4,21,22,55,24,54,23,26)(8,32,14,44,33,61,41,60)(9,35,36,63,37,56,42,38)(15,45,46,57,20,53,51,47)(31,58,43,64,34,62,52,59)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381435);