#273272

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	#3520	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

772 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)(3,10,5,8)(4,9)(6,12)(7,11)(13,22,16,25)(14,28,19,23)(15,24,31,32)(17,27)(18,26)(20,30)(21,29)(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);(1,3,7,5)(2,8,11,10)(4,15,17,31)(6,19,21,14)(9,24,27,32)(12,28,29,23)(13,20,16,18)(22,30,25,26)(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);(1,4,7,17)(2,9,11,27)(3,13,5,16)(6,20,21,18)(8,22,10,25)(12,30,29,26)(14,31,19,15)(23,32,28,24)(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);(1,5,7,3)(2,10,11,8)(4,16,17,13)(6,19,21,14)(9,25,27,22)(12,28,29,23)(15,18,31,20)(24,26,32,30)(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=hi,d^2=h,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, 180468);