#486695

order	512
nilpotency class	4
isoclinism	isoclinic class of #486694
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18203
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#62284	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

664 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250762, #72250797, #72250809, …
maximal normal subgroup of	#72250762, #72250797, #72250809, …
maximal quotient of	#72250955

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393891);