#147851

order	512
nilpotency class	4
isoclinism	isoclinic class of #147850
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

1029 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)(3,13)(4,9)(5,22,23,10)(6,11)(7,12)(8,15,30,47)(14,43)(16,41)(17,42)(18,49)(19,35,51,32)(20,33)(21,34)(24,53,54,36)(25,57)(26,38)(27,39)(28,44,58,63)(29,46,37,55)(31,50,59,64)(40,45)(48,61)(52,62,56,60);(1,3)(2,8)(4,18)(5,15)(6,25)(7,16)(9,31)(10,13)(11,37)(12,30)(14,20)(17,27)(19,50)(21,48)(22,41)(23,47)(24,55)(26,45)(28,33)(29,39)(32,49)(34,59)(35,61)(36,57)(38,58)(40,62)(42,53)(43,60)(44,52)(46,54)(51,64)(56,63);(1,4,7,21)(2,9,12,34)(3,14,16,45)(5,19,23,51)(6,26,27,20)(8,28,30,58)(10,35,22,32)(11,38,39,33)(13,40,41,43)(15,44,47,63)(17,48,25,18)(24,56,54,52)(29,59,37,31)(36,60,53,62)(42,49,57,61)(46,64,55,50)

Polycyclic presentation:

2^9|a^2=g,c^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,e^b=ehi,f^b=fi,g^b=gh,d^c=dh,f^c=fi,e^d=ei,g^d=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280779, #72282605, #72282627
maximal normal subgroup of	#72280779, #72282605, #72282627

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55047);