#366004

order	512
nilpotency class	3
isoclinism	isoclinic class of #366002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13984
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1083 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273200);