#147862

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

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=i,f^2=i,g^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, 55058);