#10086336

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,7)(3,8,10,5)(4,9,12,15)(11,13,16,14)(17,18)(19,24)(20,23)(21,25)(22,30)(26,31)(27,28)(29,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(2,7)(3,10)(9,15)(11,16)(17,19,23,27)(18,24,20,28)(21,31,32,22)(25,26,29,30)(35,41)(38,45)(42,47)(46,48);(4,12)(9,15)(11,16)(13,14)(18,20)(19,27)(21,32)(26,30)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,13)(12,16)(14,15)(17,20,23,18)(19,24,27,28)(21,29,32,25)(22,30,31,26)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,6,12)(2,9,7,15)(3,11,10,16)(5,14,8,13)(17,21,23,32)(18,25,20,29)(19,22,27,31)(24,30,28,26)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,5,6,8)(2,3,7,10)(4,13,12,14)(9,16,15,11)(17,22)(18,26)(19,21)(20,30)(23,31)(24,29)(25,28)(27,32)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9993532);