#471393

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16795
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

643 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)(5,27,30,40)(6,13)(7,12)(8,14)(9,18)(10,24)(11,29,39,28)(16,35)(17,46,48,57)(19,33)(20,34)(22,31)(23,53,56,36)(25,38)(26,37)(32,50,42,49)(41,45)(43,60,47,58)(44,59)(51,55,62,54)(52,61,63,64);(1,3)(2,9,14,15)(4,22)(5,17)(6,20)(7,19)(8,18)(10,35,38,59)(11,32,39,42)(12,34,13,33)(16,25)(21,41,37,31)(23,52)(24,44)(26,45)(27,57,40,46)(28,50)(29,49)(30,48)(36,61,53,64)(43,55)(47,54)(51,60,62,58)(56,63);(1,4,6,24,8,26,7,25)(2,10,12,21,14,38,13,37)(3,16,19,45,18,44,20,22)(5,23,28,54,30,56,29,55)(9,31,33,59,15,41,34,35)(11,36,27,51,39,53,40,62)(17,47,49,52,48,43,50,63)(32,60,46,61,42,58,57,64);(1,5,6,28,8,30,7,29)(2,11,12,27,14,39,13,40)(3,17,20,50,18,48,19,49)(4,23,24,54,26,56,25,55)(9,32,34,57,15,42,33,46)(10,36,21,51,38,53,37,62)(16,43,22,52,44,47,45,63)(31,58,35,61,41,60,59,64)

Polycyclic presentation:

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

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, 378589);