#9730337

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

3876 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)(3,6)(4,5)(7,8)(9,13)(10,16)(11,14)(12,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34)(35,36)(37,45)(38,39)(40,43)(41,47)(42,46)(44,48);(1,3,7,10)(2,6,8,16)(4,11,13,15)(5,14,9,12)(19,25)(22,29)(26,31)(30,32)(34,39)(36,40)(41,45)(44,46);(2,8)(3,10)(4,13)(12,14)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,35,38,43)(34,40,39,36)(37,42,47,48)(41,46,45,44);(1,4)(2,9)(3,11)(5,8)(6,12)(7,13)(10,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(1,5,7,9)(2,4,8,13)(3,12,10,14)(6,15,16,11)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,36)(34,35)(37,46)(38,40)(39,43)(41,48)(42,45)(44,47);(1,6)(2,10)(3,8)(4,12)(5,11)(7,16)(9,15)(13,14)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,37,38,47)(34,41,39,45)(35,42,43,48)(36,44,40,46)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9637533);