#8739433

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3806 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)(3,12)(4,9)(5,16)(6,11)(7,14)(8,15)(10,13)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,3)(2,7)(4,5)(6,14)(8,16)(9,10)(11,12)(13,15)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(34,38)(39,41)(40,45)(46,47);(1,4,6,15)(2,8,11,9)(3,5,14,13)(7,16,12,10)(17,19,23,21)(18,25,20,24)(22,27,30,29)(26,32,28,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4)(2,9)(3,13)(5,14)(6,15)(7,16)(8,11)(10,12)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,5,6,13)(2,10,11,16)(3,4,14,15)(7,9,12,8)(17,21,23,19)(18,24,20,25)(22,29,30,27)(26,31,28,32);(1,2)(3,7)(4,8)(5,16)(6,11)(9,15)(10,13)(12,14)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=gh,e^2=h,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, 8646629);