#148932

order	512
nilpotency class	3
isoclinism	isoclinic class of #2048
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#394
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11255
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

451 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:

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

Polycyclic presentation:

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

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, 56128);