#8406860

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3866 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8314056);