#147853

order	512
nilpotency class	4
isoclinism	isoclinic class of #147850
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

1021 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,e^b=ehi,f^b=fi,g^b=gh,d^c=dh,f^c=fi,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, 55049);