#478694

order	512
nilpotency class	3
isoclinism	isoclinic class of #477835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2793
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17162
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

536 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)(5,11,30,42)(6,13)(7,12)(8,14)(9,18)(10,24)(16,31)(17,43,50,61)(19,34)(20,33)(22,51)(23,53,54,39)(25,38)(26,37)(27,56,55,36)(28,41,29,40)(32,49,62,48)(35,45)(44,64,57,58)(46,59)(47,60,52,63);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,24,16,26,45,25,46)(5,17,28,48,30,50,29,49)(10,35,21,31,38,51,37,59)(11,32,40,43,42,62,41,61)(23,47,27,57,54,52,55,44)(36,60,39,64,56,63,53,58);(1,4,7,24,8,26,6,25)(2,10,13,21,14,38,12,37)(3,16,19,45,20,46,18,22)(5,27,29,54,30,55,28,23)(9,31,33,51,34,59,15,35)(11,39,41,56,42,53,40,36)(17,47,49,57,50,52,48,44)(32,60,61,64,62,63,43,58);(1,5,6,28,8,30,7,29)(2,11,12,40,14,42,13,41)(3,17,18,48,20,50,19,49)(4,23,24,27,26,54,25,55)(9,32,15,43,34,62,33,61)(10,36,21,39,38,56,37,53)(16,44,45,47,46,57,22,52)(31,58,51,60,59,64,35,63)

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=f,d^2=e,e^2=i,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,d^b=def,d^c=de,e^c=ei,g^c=gi,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, 385890);