#467982

order	512
nilpotency class	3
isoclinism	isoclinic class of #455563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16597
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

580 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=f,d^2=f,e^2=i,f^2=i,h^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,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 375178);