#485591

order	512
nilpotency class	4
isoclinism	isoclinic class of #485590
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	#12119	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

652 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)(3,15)(4,21)(5,25)(6,13)(7,11)(8,14)(9,17)(10,23)(12,28)(16,45)(18,38)(19,43)(20,44)(22,39)(24,34)(26,41)(27,37)(29,36)(30,40)(31,59)(32,51)(33,49)(35,55)(42,47)(46,52)(48,63)(50,61)(53,57)(54,62)(56,58)(60,64);(1,3,8,20)(2,9,14,33)(4,16,24,48)(5,26,27,51)(6,19,7,18)(10,31,36,62)(11,32,13,41)(12,38,40,43)(15,37,44,25)(17,30,49,28)(21,52,34,60)(22,54,55,59)(23,50,29,47)(35,63,39,45)(42,53,61,58)(46,56,64,57);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,24)(9,31)(11,34)(12,39)(13,21)(14,36)(15,42)(17,46)(18,50)(19,47)(20,48)(25,58)(26,54)(27,55)(28,57)(30,56)(32,60)(33,62)(35,40)(37,53)(38,45)(41,52)(43,63)(44,61)(49,64)(51,59);(2,11,14,13)(5,27)(9,32,33,41)(10,34,36,21)(12,25,40,37)(15,43,44,38)(17,49)(22,55)(26,51)(28,30)(31,60,62,52)(35,53,39,58)(42,63,61,45)(46,64)(54,59)(56,57)

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250033, #72250797, #72281886, …
maximal normal subgroup of	#72250033, #72250797, #72281886, …
maximal quotient of	#72250331, #72256999

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392787);