#363971

order	512
nilpotency class	3
isoclinism	isoclinic class of #363969
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13956
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2441 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,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, 271167);