#8406645

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,e^2=g,f^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, 8313841);