#8912209

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

4260 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8819405);