#468841

order	512
nilpotency class	3
isoclinism	isoclinic class of #455576
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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1043 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:

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

Polycyclic presentation:

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

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, 376037);