#128066

order	512
nilpotency class	4
isoclinism	isoclinic class of #128065
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8101
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

1249 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,d^b=dh,e^b=egi,g^b=gi,d^c=di,e^c=ei,e^d=ei,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35262);