#453845

order	512
nilpotency class	3
isoclinism	isoclinic class of #427666
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16149
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

687 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)(3,9,17,33)(4,19)(5,25)(6,14)(7,13)(8,12)(10,23)(11,26)(15,34,43,58)(16,38,45,31)(18,41,29,32)(20,30,46,48)(21,49)(22,36)(24,37)(27,39)(28,40)(35,53)(42,60,63,64)(44,57,47,59)(50,56,51,62)(52,55)(54,61);(1,3,7,29,8,17,6,18)(2,9,14,41,12,33,13,32)(4,20,23,43,22,46,24,15)(5,16,28,47,26,45,27,44)(10,34,19,48,37,58,36,30)(11,38,40,57,25,31,39,59)(21,51,52,63,54,50,53,42)(35,60,61,56,55,64,49,62);(1,4)(2,10)(3,15)(5,21)(6,24)(7,23)(8,22)(9,30)(11,35)(12,37)(13,36)(14,19)(16,42)(17,43)(18,46)(20,29)(25,55)(26,54)(27,53)(28,52)(31,56)(32,58)(33,48)(34,41)(38,62)(39,49)(40,61)(44,50)(45,63)(47,51)(57,60)(59,64);(1,5,6,28,8,26,7,27)(2,11,14,39,12,25,13,40)(3,16,29,44,17,45,18,47)(4,21,24,52,22,54,23,53)(9,31,32,59,33,38,41,57)(10,35,19,49,37,55,36,61)(15,42,20,50,43,63,46,51)(30,56,58,64,48,62,34,60)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 361041);