#146959

order	512
nilpotency class	3
isoclinism	isoclinic class of #11000
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#390
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10833
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265417, #72265430, #72265447
maximal normal subgroup of	#72265417, #72265430, #72265447

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54155);