#8749609

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=g,e^2=h,f^2=gi,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, 8656805);