#469898

order	512
nilpotency class	3
isoclinism	isoclinic class of #459756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2761
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16667
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

737 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=f,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,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, 377094);