#461725

order	512
nilpotency class	3
isoclinism	isoclinic class of #455198
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2620
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16400
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

784 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 368921);