#147843

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

1037 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)(3,13,17,37)(4,9)(5,22,24,10)(6,26)(7,12)(8,15)(11,29)(14,42,45,59)(16,47,27,43)(18,49,48,62)(19,38,52,34)(20,41)(21,36)(23,31)(25,39,57,54)(28,35)(30,55)(32,56)(33,50)(40,46)(44,60)(51,61)(53,63,58,64);(1,3)(2,8)(4,18)(5,23)(6,27)(7,17)(9,33)(10,37)(11,40)(12,31)(13,22)(14,20)(15,24)(16,29)(19,51)(21,48)(25,46)(26,32)(28,45)(30,35)(34,62)(36,61)(38,49)(39,43)(41,60)(42,63)(44,58)(47,54)(50,52)(53,55)(56,57)(59,64);(1,4,7,21)(2,9,12,36)(3,14,17,45)(5,19,24,52)(6,28,29,20)(8,30,31,60)(10,38,22,34)(11,41,26,35)(13,42,37,59)(15,44,23,55)(16,48,27,18)(25,58,57,53)(32,61,40,33)(39,63,54,64)(43,49,47,62)(46,51,56,50)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,f^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.

Relationships

maximal subgroup of	#72280771, #72282605, #72282627
maximal normal subgroup of	#72280771, #72282605, #72282627

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55039);