#9947483

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	#55163	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4104 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)(3,8,11,7)(5,9,13,12)(10,15,16,14)(17,18)(19,28)(20,21)(22,32)(23,27)(24,30)(25,31)(26,29)(33,34,40,42)(35,36,37,38)(39,47,48,41)(43,46,45,44);(1,3,6,11)(2,7,4,8)(5,10,13,16)(9,14,12,15)(17,19)(18,24)(20,22)(21,26)(23,30)(25,32)(27,28)(29,31)(33,35)(34,36)(37,40)(38,42)(39,43)(41,44)(45,48)(46,47);(17,20,23,31)(18,25,27,21)(19,22,30,29)(24,32,28,26)(33,36,40,38)(34,35,42,37)(39,44,48,46)(41,43,47,45);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,20)(18,21)(19,29)(22,30)(23,31)(24,32)(25,27)(26,28)(33,37,40,35)(34,36,42,38)(39,45,48,43)(41,44,47,46);(1,4,6,2)(3,7,11,8)(5,9,13,12)(10,15,16,14)(17,21)(18,20)(19,26)(22,24)(23,25)(27,31)(28,29)(30,32)(33,38,40,36)(34,35,42,37)(39,44,48,46)(41,45,47,43);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,22,23,29)(18,26,27,32)(19,20,30,31)(21,28,25,24)(33,39)(34,41)(35,43)(36,44)(37,45)(38,46)(40,48)(42,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=i,e^2=gh,f^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, 9854679);