#461652

order	512
nilpotency class	3
isoclinism	isoclinic class of #454981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2619
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16398
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

927 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^c=hi,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, 368848);