#485589

order	512
nilpotency class	4
isoclinism	isoclinic class of #485588
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18152
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#12116	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

656 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)(3,14)(4,9)(5,22)(6,10)(7,13)(8,16)(11,26)(12,25)(15,43)(17,42)(18,41)(19,33)(20,51)(21,34)(23,39)(24,35)(27,36)(28,45)(29,56)(30,48)(31,50)(32,53)(37,62)(38,47)(40,44)(46,61)(49,55)(52,57)(54,60)(58,64)(59,63);(1,3,7,17)(2,8,13,30)(4,15,19,44)(5,23,24,56)(6,27,25,18)(9,28,33,59)(10,29,12,39)(11,36,38,41)(14,35,42,22)(16,47,48,26)(20,52,49,64)(21,54,50,46)(31,57,34,58)(32,60,37,61)(40,51,43,55)(45,62,63,53);(1,4)(2,9)(3,15)(5,20)(6,21)(7,19)(8,28)(10,34)(11,37)(12,31)(13,33)(14,40)(16,45)(17,44)(18,46)(22,55)(23,52)(24,49)(25,50)(26,53)(27,54)(29,58)(30,59)(32,38)(35,51)(36,61)(39,57)(41,60)(42,43)(47,62)(48,63)(56,64);(2,10,13,12)(4,19)(5,24)(8,29,30,39)(9,31,33,34)(11,22,38,35)(14,41,42,36)(15,44)(16,48)(21,50)(23,56)(26,47)(28,57,59,58)(32,55,37,51)(40,61,43,60)(46,54)

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,h^a=hi,e^b=eg,f^b=fhi,g^b=gi,d^c=di,e^c=eh,e^d=ei,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392785);