#453799

order	512
nilpotency class	3
isoclinism	isoclinic class of #427841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2778
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16118
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

711 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)(3,15,19,40)(4,10,22,35)(5,24,27,38)(6,12)(7,14)(8,13)(9,18,32,29)(11,28,39,26)(16,43)(17,31)(20,50)(21,34,23,36)(25,37)(30,48)(33,52)(41,46)(42,49)(44,57,63,64)(45,58)(47,59)(51,60,56,62)(53,61)(54,55);(1,3)(2,9,13,32)(4,16,22,46)(5,25,27,49)(6,18)(7,29)(8,19)(10,30)(11,37)(12,15,14,40)(17,26,47,28)(20,51)(21,45,23,48)(24,31)(33,60,61,62)(34,43)(35,58)(36,41)(38,59)(39,42)(44,53)(50,57,55,64)(52,63)(54,56);(1,4,6,23,8,22,7,21)(2,10,14,34,13,35,12,36)(3,16,29,45,19,46,18,48)(5,20,28,52,27,54,26,53)(9,30,15,41,32,58,40,43)(11,33,38,50,39,61,24,55)(17,44,25,56,47,63,49,51)(31,57,42,60,59,64,37,62);(1,5)(2,11)(3,17)(4,20)(6,28)(7,26)(8,27)(9,31)(10,33)(12,24)(13,39)(14,38)(15,42)(16,44)(18,49)(19,47)(21,53)(22,54)(23,52)(25,29)(30,57)(32,59)(34,50)(35,61)(36,55)(37,40)(41,60)(43,62)(45,56)(46,63)(48,51)(58,64)

Polycyclic presentation:

2^9|a^2=g,b^2=efh,c^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 360995);