#10247666

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

4206 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,25)(20,24)(21,26)(22,23)(27,30)(28,32)(29,31)(33,34)(35,38)(36,39)(37,42)(40,45)(41,44)(43,47)(46,48);(2,3)(7,9)(8,10)(15,16)(18,23)(19,26)(20,28)(30,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(18,23)(19,26)(24,32)(27,29)(34,35)(39,41)(40,42)(46,47);(1,3,6,2)(4,7,12,9)(5,10,14,8)(11,15,13,16)(17,19)(18,21)(20,29)(22,26)(23,25)(24,30)(27,28)(31,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,20)(18,24)(19,27)(21,31)(22,28)(23,32)(25,30)(26,29)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21)(18,19)(20,30)(22,25)(23,26)(24,29)(27,32)(28,31)(33,37)(34,40)(35,42)(36,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154862);