#148430

order	512
nilpotency class	4
isoclinism	isoclinic class of #13391
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11123
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

669 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)(3,16)(4,23,28,46)(5,30)(6,13)(7,12)(8,35)(9,15)(10,18)(11,26,43,29)(14,32)(17,50,53,63)(19,38)(20,37)(21,47)(22,49)(24,58,55,61)(25,57,34,41)(27,44,59,42)(31,45)(33,48)(36,60,62,51)(39,56)(40,52,64,54);(1,3,7,33,8,21,32,56,9,22,31,20,6,19,5,18)(2,10,30,38,13,37,45,49,15,39,14,47,35,48,12,16)(4,24,59,54,29,60,34,53,28,55,27,52,26,51,25,17)(11,40,42,58,23,50,57,36,43,64,44,61,46,63,41,62);(1,4)(2,11)(3,17)(5,27)(6,29)(7,34)(8,26)(9,28)(10,36)(12,44)(13,46)(14,41)(15,43)(16,40)(18,55)(19,54)(20,51)(21,52)(22,53)(23,35)(24,56)(25,31)(30,57)(32,59)(33,60)(37,58)(38,63)(39,62)(42,45)(47,50)(48,61)(49,64)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,f^b=fi,d^c=dhi,e^c=ei,f^c=fhi,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, 55626);