#467981

order	512
nilpotency class	3
isoclinism	isoclinic class of #455576
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:

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

Polycyclic presentation:

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

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, 375177);