#146978

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	#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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,f^2=h,g^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	#72265582, #72265598, #72265606
maximal normal subgroup of	#72265582, #72265598, #72265606

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54174);