#9206460

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3816 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)(3,9)(4,7)(5,8)(6,11)(10,16)(12,15)(13,14)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34,38,42)(35,41,45,39)(36,46,40,37)(43,48,47,44);(1,3)(2,6)(4,12)(5,11)(7,16)(8,9)(10,14)(13,15)(21,29)(25,28)(27,30)(31,32)(33,35,38,45)(34,39,42,41)(36,47,40,43)(37,44,46,48);(1,2,5,8)(3,6,11,9)(4,7,14,13)(10,15,12,16)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,36,38,40)(34,37,42,46)(35,43,45,47)(39,48,41,44);(1,2)(3,6)(4,13)(5,8)(7,14)(9,11)(10,16)(12,15)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,37)(34,40)(35,44)(36,42)(38,46)(39,43)(41,47)(45,48);(1,4,5,14)(2,7,8,13)(3,10,11,12)(6,15,9,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,35)(34,41)(36,43)(37,44)(38,45)(39,42)(40,47)(46,48);(1,3)(2,6)(4,12)(5,11)(7,16)(8,9)(10,14)(13,15)(17,22,23,19)(18,26,20,24)(21,30,29,27)(25,32,28,31)(33,37)(34,40)(35,44)(36,42)(38,46)(39,43)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=gh,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, 9113656);