#453855

order	512
nilpotency class	3
isoclinism	isoclinic class of #427841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2778
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16158
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

687 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)(3,13,17,8)(4,18,21,48)(5,10,23,36)(6,12)(7,11)(9,20,34,55)(14,40)(15,39)(16,37,25,30)(19,49)(22,29)(24,35,26,38)(27,32)(28,42)(31,53)(33,54)(41,62,50,61)(43,46)(44,58)(45,57)(47,51)(52,60)(56,64,59,63);(1,3)(2,8)(4,14,21,43)(5,22)(6,25)(7,17)(9,31,34,28)(10,29)(11,13)(12,37)(15,23)(16,27)(18,46,48,40)(19,50,51,41)(20,53,55,42)(24,44)(26,45)(30,32)(33,59,60,56)(35,58)(36,39)(38,57)(47,62,49,61)(52,63,54,64);(1,4,6,20,7,21,27,55)(2,9,32,48,11,34,12,18)(3,14,16,42,17,43,25,53)(5,19,26,54,23,51,24,52)(8,28,37,40,13,31,30,46)(10,33,35,49,36,60,38,47)(15,41,45,64,22,50,44,63)(29,56,57,61,39,59,58,62);(1,5,6,26,7,23,27,24)(2,10,32,35,11,36,12,38)(3,15,16,45,17,22,25,44)(4,19,20,54,21,51,55,52)(8,29,37,57,13,39,30,58)(9,33,48,49,34,60,18,47)(14,41,42,64,43,50,53,63)(28,56,40,61,31,59,46,62)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361051);