#146845

order	512
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#381
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10797
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

489 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54041);