#416316

order	512
nilpotency class	3
isoclinism	isoclinic class of #396100
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2414
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15221
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

697 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)(3,14,18,32)(4,9)(5,25)(6,11)(7,13)(8,17,31,19)(10,26)(12,28)(15,33,43,40)(16,30,45,57)(20,29,47,56)(21,49)(22,35)(23,36)(24,37)(27,38)(34,50)(39,54)(41,46,53,44)(42,62,63,55)(48,58,64,61)(51,60)(52,59);(1,3,7,18)(2,8)(4,20,24,47)(5,16,27,45)(6,17,28,19)(9,33)(10,30)(11,14)(12,32)(13,31)(15,22,43,23)(21,48,52,64)(25,53)(26,44,54,46)(29,35)(34,58)(36,56)(37,40)(38,41)(39,57)(42,50,63,51)(49,62)(55,59)(60,61);(1,4,6,22,7,24,28,23)(2,9,11,35,13,37,12,36)(3,15,17,20,18,43,19,47)(5,21,26,50,27,52,54,51)(8,29,14,40,31,56,32,33)(10,34,38,59,39,60,25,49)(16,42,44,48,45,63,46,64)(30,55,41,61,57,62,53,58);(1,5,6,26,7,27,28,54)(2,10,12,25,13,39,11,38)(3,16,17,44,18,45,19,46)(4,21,22,50,24,52,23,51)(8,30,32,53,31,57,14,41)(9,34,36,49,37,60,35,59)(15,42,20,48,43,63,47,64)(29,55,33,58,56,62,40,61)

Polycyclic presentation:

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

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, 323512);