#8734926

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3887 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=g,e^2=h,f^2=g,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, 8642122);