#148309

order	512
nilpotency class	4
isoclinism	isoclinic class of #135851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1854
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11095
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

649 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:

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

Polycyclic presentation:

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

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, 55505);