#463539

order	512
nilpotency class	3
isoclinism	isoclinic class of #455690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2825
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16476
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

880 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)(4,21)(5,27,30,40)(6,13)(7,12)(8,14)(9,18)(10,25)(11,29,41,31)(16,36)(17,47,51,33)(19,35)(20,34)(22,32)(23,39,54,60)(24,38)(26,42)(28,37,55,53)(43,46)(44,50,59,49)(45,62,63,56)(48,58,64,61)(52,57);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,36)(11,33)(12,34)(13,15)(14,35)(16,24)(21,43)(23,48)(25,52)(26,46)(27,44)(28,45)(29,50)(30,51)(31,49)(32,38)(37,58)(39,56)(40,59)(41,47)(42,57)(53,61)(54,64)(55,63)(60,62);(1,4,6,26,8,25,7,24)(2,10,12,42,14,21,13,38)(3,16,18,52,20,46,19,22)(5,28,29,23,30,55,31,54)(9,32,15,43,35,57,34,36)(11,39,40,37,41,60,27,53)(17,48,49,45,51,64,50,63)(33,58,44,56,47,61,59,62);(1,5)(2,11)(3,17)(4,23,25,54)(6,29)(7,31)(8,30)(9,33)(10,37,21,53)(12,40)(13,27)(14,41)(15,44)(16,45,46,63)(18,49)(19,50)(20,51)(22,48,52,64)(24,28,26,55)(32,56,57,62)(34,59)(35,47)(36,58,43,61)(38,39,42,60)

Polycyclic presentation:

2^9|a^2=efg,c^2=ef,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,g^d=gi,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, 370735);