#253487

order	512
nilpotency class	3
isoclinism	isoclinic class of #253459
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160683);