#405019

order	512
nilpotency class	3
isoclinism	isoclinic class of #396612
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2850
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14908
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

648 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312215);