#148565

order	512
nilpotency class	4
isoclinism	isoclinic class of #13417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2041
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11189
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

641 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)(3,14,17,33)(4,19)(5,25)(6,29)(7,13)(8,16,34,49)(9,21)(10,27)(11,41)(12,31)(15,45,47,62)(18,40,30,42)(20,54,46,36)(22,39)(23,60)(24,37)(26,44,50,35)(28,43)(32,48,51,55)(38,59)(52,58)(53,57)(56,63,61,64);(1,3)(2,8)(4,20)(5,26)(6,30)(7,17)(9,36)(10,40)(11,42)(12,44)(13,34)(14,43)(15,22)(16,41)(18,31)(19,51)(21,47)(23,61)(24,46)(25,33)(27,49)(28,50)(29,35)(32,37)(38,64)(39,54)(45,60)(48,58)(52,63)(53,62)(55,59)(56,57);(1,4,5,21,6,23,27,58,7,24,28,22,31,57,41,59)(2,9,10,37,12,38,25,60,13,39,11,19,29,52,43,53)(3,15,16,46,18,48,50,61,17,47,49,20,30,55,26,56)(8,32,14,36,35,62,42,63,34,51,33,54,44,45,40,64)

Polycyclic presentation:

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

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, 55761);