#148625

order	512
nilpotency class	4
isoclinism	isoclinic class of #13391
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11199
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

627 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)(3,15,19,35)(4,20)(5,11)(6,27)(7,13)(8,14)(9,17,36,25)(10,23)(12,30)(16,47,49,64)(18,34,28,41)(21,55,50,61)(22,53)(24,54)(26,44)(29,38)(31,43)(32,42)(33,56,62,52)(37,48,45,59)(39,60)(40,57)(46,58,63,51);(1,3)(2,9)(4,21)(5,25)(6,28)(7,17)(8,19)(10,37)(11,35)(12,41)(13,15)(14,36)(16,22)(18,31)(20,47)(23,52)(24,50)(26,51)(27,46)(29,59)(30,58)(32,49)(33,39)(34,44)(38,62)(40,45)(42,55)(43,63)(48,60)(53,61)(54,64)(56,57);(1,4,5,22,8,24,7,32)(2,10,11,38,14,40,13,39)(3,16,17,21,19,49,25,50)(6,29,26,57,31,60,30,23)(9,33,15,45,36,62,35,37)(12,42,27,20,44,53,43,54)(18,52,51,48,28,56,58,59)(34,47,46,55,41,64,63,61)

Polycyclic presentation:

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

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, 55821);