#148312

order	512
nilpotency class	4
isoclinism	isoclinic class of #13502
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2077
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)(3,15,19,44)(4,21)(5,26)(6,14)(7,12)(8,13)(9,17,34,51)(10,24)(11,29)(16,46,22,56)(18,45,20,43)(23,38)(25,41)(27,36)(28,40)(30,39)(31,37)(32,60,62,64)(33,52,35,50)(42,53,61,48)(47,54,49,63)(55,57)(58,59);(1,3,5,17,7,20,29,52,8,19,30,51,6,18,28,50)(2,9,11,15,14,35,26,45,13,34,40,44,12,33,39,43)(4,22,23,49,24,53,57,60,25,16,27,47,31,48,59,64)(10,32,36,61,21,54,55,56,37,62,38,42,41,63,58,46);(1,4)(2,10)(3,16)(5,27)(6,31)(7,24)(8,25)(9,32)(11,38)(12,41)(13,37)(14,21)(15,42)(17,49)(18,53)(19,22)(20,48)(23,30)(26,58)(28,57)(29,59)(33,63)(34,62)(35,54)(36,40)(39,55)(43,56)(44,61)(45,46)(47,51)(50,64)(52,60)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^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,e^c=ei

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, 55508);