#469110

order	512
nilpotency class	3
isoclinism	isoclinic class of #455576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16645
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

659 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)(3,14)(4,20,23,36)(5,24,27,11)(6,28)(7,12)(8,13)(9,17)(10,22,37,29)(15,34,46,31)(16,42,49,59)(18,33)(19,43)(21,38)(25,35)(26,39,30,40)(32,48,55,51)(41,45,52,50)(44,57)(47,62)(53,61)(54,56)(58,63)(60,64);(1,3,7,18,8,19,6,17)(2,9,28,43,13,33,12,14)(4,15,22,45,23,46,29,50)(5,16,26,48,27,49,30,51)(10,34,36,41,37,31,20,52)(11,32,39,59,24,55,40,42)(21,44,53,63,54,64,25,47)(35,60,56,58,61,57,38,62);(1,4)(2,10,13,37)(3,15)(5,25)(6,29)(7,22)(8,23)(9,31,33,34)(11,38,24,56)(12,20,28,36)(14,41,43,52)(16,47)(17,50)(18,45)(19,46)(21,26)(27,53)(30,54)(32,58,55,62)(35,40,61,39)(42,60,59,57)(44,48)(49,63)(51,64);(1,5,7,26,8,27,6,30)(2,11,12,40,13,24,28,39)(3,16,18,48,19,49,17,51)(4,21,22,53,23,54,29,25)(9,32,14,42,33,55,43,59)(10,35,20,38,37,61,36,56)(15,44,45,63,46,64,50,47)(31,57,41,58,34,60,52,62)

Polycyclic presentation:

2^9|a^2=efg,b^2=fg,c^2=g,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,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, 376306);