#148445

order	512
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#397
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11147
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

637 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=di,g^b=gi,f^c=fi,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, 55641);