#9159924

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	#40736	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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)(3,10,5,7)(4,9,14,13)(11,16,12,15)(17,18,22,23)(19,24,27,30)(20,25,31,28)(21,29,32,26)(33,34,40,37)(35,41,45,39)(36,38,42,47)(43,48,46,44);(1,3,6,5)(2,7,8,10)(4,12,14,11)(9,16,13,15)(17,18,22,23)(19,24,27,30)(20,28,31,25)(21,26,32,29)(33,35,40,45)(34,41,37,39)(36,46,42,43)(38,44,47,48);(2,8)(7,10)(9,13)(15,16)(18,23)(19,27)(25,28)(26,29)(33,36,40,42)(34,38,37,47)(35,43,45,46)(39,44,41,48);(1,2)(3,7)(4,13)(5,10)(6,8)(9,14)(11,16)(12,15)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,32)(26,31)(33,37,40,34)(35,39,45,41)(36,38,42,47)(43,48,46,44);(1,4,6,14)(2,9,8,13)(3,11,5,12)(7,15,10,16)(17,20)(18,25)(19,26)(21,24)(22,31)(23,28)(27,29)(30,32)(33,38)(34,42)(35,44)(36,37)(39,46)(40,47)(41,43)(45,48);(1,5,6,3)(2,10,8,7)(4,11,14,12)(9,15,13,16)(17,21)(18,26)(19,28)(20,30)(22,32)(23,29)(24,31)(25,27)(33,39)(34,35)(36,44)(37,45)(38,43)(40,41)(42,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067120);