#436614

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15566
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

911 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343810);