#9203652

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

3885 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)(3,7)(4,5)(6,9)(8,10)(11,12)(13,16)(14,15)(17,18)(19,25)(20,24)(21,23)(22,26)(27,32)(28,31)(29,30)(33,34,39,36)(35,42,38,40)(37,41,45,44)(43,48,46,47);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,14)(15,16)(17,19,22,21)(18,23,26,25)(20,28,30,27)(24,32,29,31)(37,45)(41,44)(43,46)(47,48);(3,10)(7,8)(11,13)(12,16)(17,18,22,26)(19,23,21,25)(20,24,30,29)(27,31,28,32)(33,35,39,38)(34,40,36,42)(37,43,45,46)(41,47,44,48);(1,3)(2,8)(4,13)(5,12)(6,10)(7,9)(11,14)(15,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,30)(27,32)(28,31)(33,36,39,34)(35,42,38,40)(37,44,45,41)(43,48,46,47);(1,4,6,14)(2,5,9,15)(3,11,10,13)(7,12,8,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,32)(26,29)(33,37,39,45)(34,41,36,44)(35,43,38,46)(40,47,42,48);(1,5)(2,4)(3,12)(6,15)(7,11)(8,13)(9,14)(10,16)(17,21,22,19)(18,25,26,23)(20,27,30,28)(24,31,29,32)(33,38,39,35)(34,42,36,40)(37,46,45,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,e^2=ghi,f^2=h,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, 9110848);