#8943749

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	#42974	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8850945);