#147993

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#394
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11042
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

665 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55189);