#471101

order	512
nilpotency class	3
isoclinism	isoclinic class of #459145
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16750
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

575 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)(3,9,17,33)(4,19)(5,25,27,11)(6,14)(7,13)(8,12)(10,22)(15,34,43,60)(16,44)(18,40,28,32)(20,30,51,56)(21,49,53,61)(23,36)(24,37)(26,39,29,41)(31,46)(35,54,38,52)(42,59)(45,62)(47,57)(48,58)(50,55)(63,64);(1,3)(2,9,12,33)(4,20,24,51)(5,16)(6,28)(7,18)(8,17)(10,34)(11,31,25,44)(13,40,14,32)(15,23,43,22)(19,30)(21,50,53,63)(26,48)(27,46)(29,47)(35,59)(36,60)(37,56)(38,62)(39,58,41,57)(42,54,45,52)(49,55)(61,64);(1,4)(2,10,12,36)(3,15,17,43)(5,21)(6,22)(7,23)(8,24)(9,30)(11,38,25,35)(13,19,14,37)(16,45,46,42)(18,20,28,51)(26,52)(27,53)(29,54)(31,55)(32,34)(33,56)(39,61,41,49)(40,60)(44,64)(47,63,48,50)(57,59)(58,62);(1,5,6,26,8,27,7,29)(2,11,14,41,12,25,13,39)(3,16,28,48,17,46,18,47)(4,21,22,52,24,53,23,54)(9,31,32,57,33,44,40,58)(10,35,37,61,36,38,19,49)(15,42,51,63,43,45,20,50)(30,55,34,59,56,64,60,62)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=e,d^2=ef,f^2=i,g^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,e^b=ei,f^b=fi,d^c=de,h^c=hi,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, 378297);