#10218697

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	#40598	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10125893);