#9206423

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	#2828	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3801 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:

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

Polycyclic presentation:

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

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, 9113619);