#8722283

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8629479);