#436223

order	512
nilpotency class	3
isoclinism	isoclinic class of #436222
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15551
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

733 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)(3,15)(4,21,26,37)(5,27)(6,14)(7,12)(8,13)(9,18)(10,25,38,24)(11,29)(16,43,22,32)(17,47)(19,34)(20,35)(23,53,57,36)(28,39)(30,40)(31,41)(33,49)(42,46,50,45)(44,64,52,61)(48,60)(51,56,62,55)(54,59,58,63);(1,3,8,20)(2,9,13,34)(4,22,26,16)(5,28,31,48)(6,19,7,18)(10,32,38,43)(11,39,40,60)(12,35,14,15)(17,30,49,29)(21,50,37,42)(23,54,57,58)(24,45,25,46)(27,33,41,47)(36,61,53,64)(44,55,52,56)(51,59,62,63);(1,4,6,24,8,26,7,25)(2,10,12,21,13,38,14,37)(3,16,18,45,20,22,19,46)(5,23,29,55,31,57,30,56)(9,32,15,42,34,43,35,50)(11,36,27,51,40,53,41,62)(17,44,28,58,49,52,48,54)(33,59,39,61,47,63,60,64);(1,5,7,30,8,31,6,29)(2,11,12,27,13,40,14,41)(3,17,19,48,20,49,18,28)(4,23,25,56,26,57,24,55)(9,33,15,39,34,47,35,60)(10,36,21,51,38,53,37,62)(16,44,46,54,22,52,45,58)(32,59,42,61,43,63,50,64)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343419);