#478693

order	512
nilpotency class	3
isoclinism	isoclinic class of #478580
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2789
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17162
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

541 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)(5,27,32,44)(6,13)(7,12)(8,14)(9,18)(10,25)(11,31,43,30)(16,46)(17,41,28,34)(19,36)(20,35)(22,37)(23,38,56,62)(24,40)(26,39)(29,53,55,42)(33,49)(45,52,57,51)(47,61,64,60)(48,59)(50,63,54,58);(1,3,6,18,8,20,7,19)(2,9,12,15,14,36,13,35)(4,22,24,16,26,48,25,49)(5,28,30,52,32,17,31,51)(10,37,39,33,40,59,21,46)(11,41,27,57,43,34,44,45)(23,54,29,47,56,50,55,64)(38,61,42,58,62,60,53,63);(1,4,6,25,8,26,7,24)(2,10,12,21,14,40,13,39)(3,16,18,22,20,49,19,48)(5,29,30,23,32,55,31,56)(9,33,15,37,36,46,35,59)(11,42,27,38,43,53,44,62)(17,50,51,47,28,54,52,64)(34,60,45,58,41,61,57,63);(1,5,6,30,8,32,7,31)(2,11,12,27,14,43,13,44)(3,17,18,51,20,28,19,52)(4,23,24,29,26,56,25,55)(9,34,15,45,36,41,35,57)(10,38,39,42,40,62,21,53)(16,47,48,50,49,64,22,54)(33,58,59,60,46,63,37,61)

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

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, 385889);