#8649884

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

4029 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,22,23)(19,24,30,27)(20,21,26,25)(28,31,32,29)(33,34,39,37)(35,40,36,43)(38,41,46,45)(42,47,44,48);(2,7)(3,11)(4,14)(13,15)(17,18,22,23)(19,27,30,24)(20,21,26,25)(28,29,32,31)(34,37)(40,43)(41,45)(47,48);(2,7)(4,14)(8,10)(12,16)(18,23)(20,26)(24,27)(28,32)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,3)(2,8)(4,12)(5,15)(6,11)(7,10)(9,13)(14,16)(17,19)(18,24)(20,28)(21,31)(22,30)(23,27)(25,29)(26,32)(33,36,39,35)(34,40,37,43)(38,42,46,44)(41,48,45,47);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,20,22,26)(18,25,23,21)(19,28,30,32)(24,29,27,31)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,21,22,25)(18,26,23,20)(19,29,30,31)(24,28,27,32)(33,38,39,46)(34,41,37,45)(35,42,36,44)(40,47,43,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8557080);