#9947276

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	#55102	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854472);