#9160787

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4041 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067983);