#462215

order	512
nilpotency class	3
isoclinism	isoclinic class of #462214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16422
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

691 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)(3,14,18,43)(4,10,23,36)(5,24)(6,12)(7,29)(8,13)(9,17,33,19)(11,26)(15,41)(16,32,48,61)(20,51)(21,38,53,62)(22,35,28,37)(25,56,54,34)(27,39)(30,40)(31,45)(42,50,55,47)(44,58)(46,60)(49,59)(52,64)(57,63);(1,3)(2,9,13,33)(4,20,23,15)(5,16)(6,17)(7,19)(8,18)(10,31)(11,32,40,61)(12,14,29,43)(21,52,53,44)(22,49,28,45)(24,55,39,42)(25,57,54,46)(26,47)(27,48)(30,50)(34,58)(35,51)(36,59)(37,41)(38,60)(56,64)(62,63);(1,4,7,22,8,23,6,28)(2,10,12,37,13,36,29,35)(3,15,19,45,18,20,17,49)(5,25,30,53,27,54,26,21)(9,31,14,41,33,59,43,51)(11,38,39,34,40,62,24,56)(16,46,50,52,48,57,47,44)(32,60,42,58,61,63,55,64);(1,5,6,26,8,27,7,30)(2,11,12,39,13,40,29,24)(3,16,17,47,18,48,19,50)(4,21,22,25,23,53,28,54)(9,32,14,42,33,61,43,55)(10,34,35,38,36,56,37,62)(15,44,45,46,20,52,49,57)(31,58,51,60,59,64,41,63)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369411);