#273759

order	512
nilpotency class	3
isoclinism	isoclinic class of #273755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3514	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

753 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fh,g^c=gh,e^d=ei,f^d=fhi

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, 180955);