#147847

order	512
nilpotency class	4
isoclinism	isoclinic class of #147846
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10995
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3449	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1025 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,d^c=dh,f^c=fi,g^c=gi,e^d=ei,g^d=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55043);