#8712851

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

4107 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)(5,8)(6,7)(11,14)(12,13)(15,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(33,34,40,46)(35,41,37,43)(36,44,38,42)(39,45,48,47);(2,7)(3,10)(8,16)(11,13)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,35)(34,41)(36,45)(37,40)(38,47)(39,42)(43,46)(44,48);(2,7)(3,10)(8,16)(11,13)(19,28)(20,29)(24,30)(25,32)(33,36,40,38)(34,42,46,44)(35,47,37,45)(39,41,48,43);(1,3)(2,4)(5,13)(6,10)(7,9)(8,14)(11,15)(12,16)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,37,40,35)(34,43,46,41)(36,47,38,45)(39,42,48,44);(1,4)(2,3)(5,14)(6,9)(7,10)(8,13)(11,16)(12,15)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,38,40,36)(34,44,46,42)(35,47,37,45)(39,41,48,43);(1,5,6,15)(2,8,7,16)(3,11,10,13)(4,12,9,14)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,39)(34,45)(35,42)(36,41)(37,44)(38,43)(40,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8620047);