#492891

order	512
nilpotency class	4
isoclinism	isoclinic class of #33348
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	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂²×#342
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#585 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2281 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)(4,9)(5,21,22,10)(6,11)(7,12)(8,15,26,39)(14,33)(16,34)(17,35)(18,28)(19,44,42,29)(20,30)(23,46,47,31)(24,32)(25,37,48,56)(27,40,50,59)(36,53)(38,54)(41,55)(43,51)(45,61,60,52)(49,58,62,64)(57,63);(1,3,6,16,7,17,24,41)(2,8,11,27,12,26,32,50)(4,14,20,38,18,36,43,57)(5,15,23,40,22,39,47,59)(9,25,30,49,28,48,51,62)(10,13,31,34,21,35,46,55)(19,37,45,58,42,56,60,64)(29,33,52,54,44,53,61,63);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,25)(10,29)(11,30)(12,28)(13,33)(15,37)(16,38)(17,36)(21,44)(22,42)(23,45)(24,43)(26,48)(27,49)(31,52)(32,51)(34,54)(35,53)(39,56)(40,58)(41,57)(46,61)(47,60)(50,62)(55,63)(59,64);(4,18)(9,28)(14,36)(19,42)(20,43)(25,48)(29,44)(30,51)(33,53)(37,56)(38,57)(45,60)(49,62)(52,61)(54,63)(58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303182, #72303215, #72303219, …
maximal normal subgroup of	#72303182, #72303215, #72303219, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400087);