#146977

order	512
nilpotency class	3
isoclinism	isoclinic class of #11234
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#391
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10835
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

537 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265577, #72265598, #72265606
maximal normal subgroup of	#72265577, #72265598, #72265606

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54173);