#146766

order	512
nilpotency class	4
isoclinism	isoclinic class of #13030
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1879
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10788
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

939 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dhi,e^b=ehi,h^b=hi,d^c=dhi,e^c=eh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314225
maximal normal subgroup of	#72314225

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53962);