#455643

order	512
nilpotency class	3
isoclinism	isoclinic class of #455397
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16267
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

902 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,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, 362839);