#494655

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1893
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1263 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)(3,14,17,39)(4,20,23,38)(5,24)(6,12)(7,31)(8,13)(9,18,33,25)(10,22,37,30)(11,15)(16,47)(19,43,29,44)(21,36,26,40)(27,41)(28,57)(32,58)(34,53,35,45)(42,51)(46,62,56,55)(48,63)(49,61)(50,60)(52,54)(59,64);(1,3,15,45,6,29,5,25,8,17,27,53,7,19,28,18)(2,9,24,43,31,34,11,39,13,33,57,44,12,35,41,14)(4,16,46,58,30,50,26,49,23,48,56,64,22,51,21,52)(10,32,55,63,20,54,40,60,37,59,62,47,38,61,36,42);(1,4)(2,10)(3,16)(5,26)(6,30)(7,22)(8,23)(9,32)(11,40)(12,38)(13,37)(14,42)(15,46)(17,48)(18,52)(19,51)(20,31)(21,28)(24,55)(25,49)(27,56)(29,50)(33,59)(34,54)(35,61)(36,41)(39,60)(43,63)(44,47)(45,58)(53,64)(57,62);(3,17)(9,33)(14,39)(16,48)(18,25)(19,29)(32,59)(34,35)(42,60)(43,44)(45,53)(47,63)(49,52)(50,51)(54,61)(58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401851);