#147748

order	512
nilpotency class	4
isoclinism	isoclinic class of #147745
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10980
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1020 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:

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

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,e^a=eh,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,f^c=fi,g^c=gh,g^d=gi,g^e=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, 54944);