#8926309

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

3986 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)(3,9)(4,10)(5,13)(6,8)(7,12)(11,15)(14,16)(19,25)(24,31)(26,29)(27,32)(34,38)(39,44)(40,45)(46,48);(1,3)(2,7)(4,11)(5,10)(6,12)(8,9)(13,14)(15,16)(17,18,22,23)(19,24,25,31)(20,28,30,21)(26,27,29,32)(36,43)(40,45)(41,47)(46,48);(2,8)(5,15)(7,9)(10,16)(18,23)(21,28)(24,31)(27,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48);(1,3)(2,9)(4,13)(5,10)(6,12)(7,8)(11,14)(15,16)(17,19)(18,24)(20,29)(21,27)(22,25)(23,31)(26,30)(28,32)(33,35,37,42)(34,39,38,44)(36,41,43,47)(40,46,45,48);(1,4,6,14)(2,10,8,16)(3,11,12,13)(5,9,15,7)(17,20)(18,21)(19,26)(22,30)(23,28)(24,32)(25,29)(27,31)(33,36)(34,40)(35,41)(37,43)(38,45)(39,46)(42,47)(44,48);(1,5,6,15)(2,11,8,13)(3,10,12,16)(4,9,14,7)(17,21)(18,20)(19,27)(22,28)(23,30)(24,29)(25,32)(26,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48)

Polycyclic presentation:

2^9|b^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833505);