#8919496

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

4043 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=gh,e^2=hi,f^2=h,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, 8826692);