#494451

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1874
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

1347 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401647);