#492911

order	512
nilpotency class	4
isoclinism	isoclinic class of #33359
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#99
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂²×#342
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#585 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2261 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)(3,13,16,34)(4,9)(5,21,22,10)(6,24)(7,12)(8,15)(11,25)(14,33,36,55)(17,42,41,35)(18,29)(19,45,43,30)(20,47)(23,32,49,48)(26,37)(27,53)(28,40)(31,44)(38,60,58,56)(39,50)(46,54,61,62)(51,57)(52,64)(59,63);(1,3,6,17,7,16,25,41)(2,8,11,27,12,28,24,50)(4,14,20,38,18,36,44,58)(5,15,23,39,22,40,49,53)(9,26,31,52,29,51,47,63)(10,13,32,35,21,34,48,42)(19,37,46,59,43,57,61,64)(30,33,54,56,45,55,62,60);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,26)(10,30)(11,31)(12,29)(13,33)(15,37)(16,36)(17,38)(21,45)(22,43)(23,46)(24,47)(25,44)(27,52)(28,51)(32,54)(34,55)(35,56)(39,59)(40,57)(41,58)(42,60)(48,62)(49,61)(50,63)(53,64);(4,18)(9,29)(14,36)(19,43)(20,44)(26,51)(30,45)(31,47)(33,55)(37,57)(38,58)(46,61)(52,63)(54,62)(56,60)(59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303182, #72303191, #72303219, …
maximal normal subgroup of	#72303182, #72303191, #72303219, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400107);