#9713742

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	#55414	C₂
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)(3,9)(4,5)(6,8)(7,13)(10,15)(11,14)(12,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,32)(28,31)(29,30)(33,34)(35,43)(36,46)(37,40)(38,42)(39,41)(44,47)(45,48);(1,3,6,13)(2,7,8,9)(4,11,10,16)(5,12,15,14)(17,19)(18,22)(20,26)(21,29)(23,25)(24,30)(27,31)(28,32)(33,35,40,39)(34,41,37,43)(36,45,38,44)(42,48,46,47);(2,8)(4,10)(7,9)(11,16)(18,20)(19,25)(24,28)(27,29)(33,36,40,38)(34,42,37,46)(35,44,39,45)(41,47,43,48);(1,4,6,10)(2,5,8,15)(3,11,13,16)(7,12,9,14)(17,20,23,18)(19,26,25,22)(21,28,31,24)(27,30,29,32)(33,37,40,34)(35,43,39,41)(36,42,38,46)(44,47,45,48);(1,3)(2,9)(4,11)(5,14)(6,13)(7,8)(10,16)(12,15)(17,21,23,31)(18,24,20,28)(19,27,25,29)(22,32,26,30)(33,38,40,36)(34,42,37,46)(35,45,39,44)(41,47,43,48);(1,5)(2,10)(3,12)(4,8)(6,15)(7,16)(9,11)(13,14)(17,22,23,26)(18,25,20,19)(21,30,31,32)(24,27,28,29)(33,39,40,35)(34,43,37,41)(36,45,38,44)(42,48,46,47)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=h,e^2=i,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, 9620938);