#8739209

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3807 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8646405);