#147849

order	512
nilpotency class	4
isoclinism	isoclinic class of #147846
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:

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

Polycyclic presentation:

2^9|a^2=g,c^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,f^c=fi,g^c=gi,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, 55045);