#8942656

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

4052 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8849852);