#147860

order	512
nilpotency class	4
isoclinism	isoclinic class of #147859
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

1019 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)(3,15)(4,20)(5,26,28,11)(6,13)(7,12)(8,14)(9,17,34,50)(10,23)(16,45)(18,51)(19,44)(21,42)(22,56,58,54)(24,38)(25,37)(27,40,29,41)(30,43)(31,60,53,55)(32,49,33,48)(35,63,62,46)(36,59,39,57)(47,64)(52,61);(1,3)(2,9)(4,21)(5,17)(6,18)(7,30)(8,19)(10,35)(11,15)(12,33)(13,32)(14,34)(16,24)(20,53)(22,55)(23,47)(25,52)(26,44)(27,48)(28,50)(29,49)(31,37)(36,64)(38,62)(39,45)(40,51)(41,43)(42,54)(46,57)(56,61)(58,60)(59,63);(1,4,8,25)(2,10,14,38)(3,16,19,47)(5,22,28,58)(6,24,7,23)(9,31,34,53)(11,39,26,36)(12,37,13,20)(15,42,44,61)(17,46,50,63)(18,52,30,21)(27,59,29,57)(32,35,33,62)(40,56,41,54)(43,64,51,45)(48,55,49,60)

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^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,e^c=ei,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, 55056);