#8864549

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	#35044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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:

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

Polycyclic presentation:

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

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, 8771745);