#9059126

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	#35811	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8966322);