#456381

order	512
nilpotency class	3
isoclinism	isoclinic class of #456381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16282
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1148 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:

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

Polycyclic presentation:

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

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, 363577);