#147845

order	512
nilpotency class	4
isoclinism	isoclinic class of #147842
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

1013 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,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, 55041);