#364614

order	512
nilpotency class	3
isoclinism	isoclinic class of #364611
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13967
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2301 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=i,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^c=hi,e^d=ei

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, 271810);