#8990297

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

4210 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)(3,11)(4,9)(5,15)(6,7)(8,12)(10,16)(13,14)(33,34,38,40)(35,41,42,36)(37,39,45,46)(43,47,48,44);(2,7)(4,14)(8,11)(10,15)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,34)(35,36)(37,39)(38,40)(41,42)(43,44)(45,46)(47,48);(2,7)(3,12)(4,14)(5,16)(18,19)(20,27)(21,29)(30,31)(35,42)(36,41)(37,45)(39,46);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,35,38,42)(34,36,40,41)(37,43,45,48)(39,44,46,47);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,47)(46,48);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,39,40,46)(35,43,42,48)(36,44,41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8897493);