#368247

order	512
nilpotency class	3
isoclinism	isoclinic class of #23996
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2404
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14016
prime radical	#3449	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1371 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317342, #72317343, #72317353
maximal normal subgroup of	#72317342, #72317343, #72317353

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275443);