#9160778

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	#40446	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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)(17,18,23,25)(19,26,22,24)(20,29,30,21)(27,32,31,28)(33,34,39,43)(35,44,48,40)(36,41,46,38)(37,42,45,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,19,23,22)(18,24,25,26)(20,27,30,31)(21,32,29,28)(33,35)(34,40)(36,45)(37,46)(38,47)(39,48)(41,42)(43,44);(2,8)(3,11)(9,16)(10,12)(18,25)(20,30)(24,26)(27,31)(33,36,39,46)(34,41,43,38)(35,45,48,37)(40,42,44,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,14)(15,16)(17,20,23,30)(18,21,25,29)(19,27,22,31)(24,32,26,28)(33,36)(34,38)(35,45)(37,48)(39,46)(40,47)(41,43)(42,44);(1,4,6,14)(2,9,8,16)(3,12,11,10)(5,7,15,13)(17,21)(18,20)(19,28)(22,32)(23,29)(24,31)(25,30)(26,27)(33,37,39,45)(34,42,43,47)(35,46,48,36)(38,40,41,44);(1,5,6,15)(2,10,8,12)(3,9,11,16)(4,13,14,7)(17,22,23,19)(18,26,25,24)(20,31,30,27)(21,28,29,32)(33,38)(34,36)(35,47)(37,44)(39,41)(40,45)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=hi,e^2=gh,f^2=h,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, 9067974);