#273267

order	512
nilpotency class	3
isoclinism	isoclinic class of #273243
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	#121	C₂×D₄
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)(3,11,14,7)(4,16,17,8)(5,10)(6,9)(12,19,23,15)(13,22)(18,29,30,24)(20,21)(25,28)(26,31)(27,32)(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);(1,3,6,14)(2,7,9,11)(4,13,17,28)(5,19,21,15)(8,22,16,25)(10,12,20,23)(18,31,30,27)(24,26,29,32)(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);(1,2)(3,12)(4,8)(5,20)(6,9)(7,19)(10,21)(11,15)(13,26)(14,23)(16,17)(18,29)(22,31)(24,30)(25,27)(28,32)(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)(2,8)(3,13)(5,18)(6,17)(7,22)(9,16)(10,24)(11,25)(12,26)(14,28)(15,27)(19,31)(20,29)(21,30)(23,32)(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,b^2=h,c^2=i,e^2=i,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,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180463);