#9206432

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3814 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)(3,4)(5,13)(6,9)(7,11)(8,15)(10,14)(12,16)(18,21)(23,26)(24,29)(30,32)(33,34,38,39)(35,42,40,36)(37,41,45,46)(43,48,47,44);(1,3,6,11)(2,7,9,4)(5,10,15,16)(8,14,13,12)(20,28)(24,29)(25,31)(30,32)(34,39)(35,40)(37,45)(44,48);(1,2,6,9)(3,7,11,4)(5,8,15,13)(10,14,16,12)(17,18,22,21)(19,23,27,26)(20,24,28,29)(25,30,31,32)(33,35,38,40)(34,36,39,42)(37,43,45,47)(41,44,46,48);(1,2)(3,7)(4,11)(5,8)(6,9)(10,14)(12,16)(13,15)(17,19,22,27)(18,23,21,26)(20,25,28,31)(24,30,29,32)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,46)(45,48);(1,4,6,7)(2,3,9,11)(5,14,15,12)(8,16,13,10)(17,20)(18,24)(19,25)(21,29)(22,28)(23,30)(26,32)(27,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,8)(3,10)(4,12)(6,15)(7,14)(9,13)(11,16)(17,21,22,18)(19,26,27,23)(20,29,28,24)(25,32,31,30)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,46)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,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, 9113628);