#368404

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2408
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14018
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1289 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275600);