#8994249

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	#47807	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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)(3,11,12,7)(4,13,14,9)(5,10,15,16)(33,34)(35,44)(36,38)(37,41)(39,43)(40,45)(42,48)(46,47);(1,3,6,12)(2,7,8,11)(4,5,14,15)(9,10,13,16)(17,18,22,26)(19,24,29,23)(20,21,25,32)(27,31,30,28)(35,45)(37,46)(40,44)(41,47);(2,8)(4,14)(5,15)(7,11)(17,19,22,29)(18,23,26,24)(20,30,25,27)(21,31,32,28)(33,35,39,45)(34,40,43,44)(36,37,42,46)(38,47,48,41);(1,4,6,14)(2,9,8,13)(3,5,12,15)(7,10,11,16)(17,20,22,25)(18,21,26,32)(19,27,29,30)(23,28,24,31)(33,36,39,42)(34,38,43,48)(35,46,45,37)(40,41,44,47);(1,5)(2,10)(3,4)(6,15)(7,9)(8,16)(11,13)(12,14)(17,19)(18,24)(20,30)(21,28)(22,29)(23,26)(25,27)(31,32)(33,37,39,46)(34,41,43,47)(35,36,45,42)(38,40,48,44);(1,2)(3,7)(4,9)(5,10)(6,8)(11,12)(13,14)(15,16)(17,21)(18,25)(19,28)(20,26)(22,32)(23,30)(24,27)(29,31)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8901445);