#146790

order	512
nilpotency class	4
isoclinism	isoclinic class of #12952
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
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

959 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)(3,14)(4,19,22,57)(5,24)(6,29)(7,33)(8,12)(9,16)(10,21,40,20)(11,28)(13,27)(15,49,51,36)(17,42)(18,37)(23,59,35,43)(25,39)(26,46,31,41)(30,45)(32,44)(34,48)(38,54)(47,50,56,60)(52,63,55,64)(53,62,58,61);(1,3)(2,9,12,38)(4,15)(5,25)(6,30)(7,34)(8,18)(10,36,40,49)(11,42,44,48)(13,45,33,39)(14,24,37,29)(16,32)(17,27)(19,56,57,47)(20,58)(21,53)(22,51)(23,55)(26,60)(28,54)(31,50)(35,52)(41,64,46,63)(43,62,59,61);(1,4)(2,10)(3,15)(5,26)(6,31)(7,35)(8,22)(9,36)(11,43)(12,40)(13,46)(14,47)(16,53)(17,55)(18,51)(19,24)(20,28)(21,32)(23,27)(25,60)(29,57)(30,50)(33,41)(34,52)(37,56)(38,49)(39,64)(42,62)(44,59)(45,63)(48,61)(54,58)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280192, #72280442, #72281094, …
maximal normal subgroup of	#72280192, #72280442, #72281094, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53986);