#468842

order	512
nilpotency class	3
isoclinism	isoclinic class of #455563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16632
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1049 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,e^2=i,f^2=i,h^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,h^b=hi,d^c=de,e^c=ei,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, 376038);