#478828

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17170
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

637 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)(4,9)(5,10,27,42)(6,29)(7,12)(8,16)(11,19)(14,34)(15,39,51,45)(17,44)(18,31)(20,53,54,35)(21,37)(22,36)(23,38)(24,32,50,61)(25,40,55,63)(26,43,28,41)(30,33)(46,62,56,60)(47,59)(48,58)(49,64,52,57);(1,3)(2,8)(4,18)(5,24)(6,30)(7,17)(9,34)(10,39)(11,44)(12,33)(13,29)(14,22)(15,28)(16,19)(20,46)(21,48)(23,47)(25,52)(26,51)(27,50)(31,37)(32,43)(35,57)(36,59)(38,58)(40,62)(41,61)(42,45)(49,55)(53,64)(54,56)(60,63);(1,4,19,21,7,23,6,22)(2,9,29,36,12,38,11,37)(3,14,30,47,17,48,16,18)(5,25,28,54,27,55,26,20)(8,31,44,58,33,59,13,34)(10,40,43,53,42,63,41,35)(15,49,24,56,51,52,50,46)(32,60,39,64,61,62,45,57);(1,5,6,26,7,27,19,28)(2,10,11,41,12,42,29,43)(3,15,16,50,17,51,30,24)(4,20,21,25,23,54,22,55)(8,32,13,45,33,61,44,39)(9,35,36,40,38,53,37,63)(14,46,47,49,48,56,18,52)(31,57,58,60,59,64,34,62)

Polycyclic presentation:

2^9|a^2=efg,c^2=ef,d^2=e,e^2=i,g^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=def,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386024);