#501532

order	512
nilpotency class	4
isoclinism	isoclinic class of #501530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24658
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

568 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408728);