#147929

order	512
nilpotency class	4
isoclinism	isoclinic class of #13323
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11022
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂²×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)(3,15,19,47)(4,21,25,10)(5,27)(6,30)(7,33)(8,12)(9,17,37,53)(11,28)(13,29)(14,32)(16,49)(18,45,54,39)(20,38,34,48)(22,36)(23,57,31,42)(24,44,59,41)(26,43,35,46)(40,52)(50,62)(51,63)(55,64)(56,60)(58,61);(1,3)(2,9)(4,22)(5,18)(6,17)(7,34)(8,19)(10,40)(11,38)(12,37)(13,15)(14,45)(16,24)(20,32)(21,49)(23,51)(25,58)(26,50)(27,47)(28,53)(29,54)(30,48)(31,56)(33,39)(35,55)(36,42)(41,62)(43,60)(44,64)(46,63)(52,59)(57,61);(1,4)(2,10)(3,16)(5,24)(6,31)(7,35)(8,25)(9,36)(11,42)(12,21)(13,44)(14,46)(15,40)(17,22)(18,55)(19,52)(20,56)(23,28)(26,32)(27,41)(29,59)(30,57)(33,43)(34,51)(37,61)(38,63)(39,64)(45,62)(47,49)(48,60)(50,54)(53,58)

Polycyclic presentation:

2^9|a^2=g,d^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=gi,h^b=hi,d^c=dh,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55125);