#9726052

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(3,8)(4,13)(5,10)(6,14)(7,9)(11,16)(12,15)(17,18)(19,25)(20,29)(21,23)(22,26)(24,27)(28,30)(31,32)(33,34,39,43)(35,44,46,40)(36,47,48,41)(37,38,42,45);(1,3,7,4)(2,8,9,13)(5,14,15,11)(6,12,16,10)(17,19,24,21)(18,25,27,23)(20,28,22,31)(26,32,29,30)(33,35)(34,40)(36,45)(37,41)(38,48)(39,46)(42,47)(43,44);(2,9)(3,4)(10,12)(11,14)(17,20,24,22)(18,26,27,29)(19,28,21,31)(23,30,25,32)(33,36,39,48)(34,41,43,47)(35,45,46,38)(37,44,42,40);(1,4,7,3)(2,8,9,13)(5,14,15,11)(6,10,16,12)(17,21,24,19)(18,23,27,25)(20,28,22,31)(26,32,29,30)(33,37,39,42)(34,38,43,45)(35,41,46,47)(36,40,48,44);(1,5,7,15)(2,10,9,12)(3,11,4,14)(6,8,16,13)(17,22,24,20)(18,26,27,29)(19,28,21,31)(23,32,25,30)(33,38)(34,42)(35,36)(37,43)(39,45)(40,41)(44,47)(46,48);(1,6,7,16)(2,11,9,14)(3,12,4,10)(5,13,15,8)(17,23)(18,19)(20,30)(21,27)(22,32)(24,25)(26,28)(29,31)(33,35)(34,40)(36,45)(37,41)(38,48)(39,46)(42,47)(43,44)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=i,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, 9633248);