#10246917

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

4022 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)(5,10)(8,14)(9,11)(12,16)(13,15)(17,18,23,22)(19,26,21,24)(20,28,30,25)(27,31,29,32)(33,34,39,36)(35,38,41,43)(37,40,47,44)(42,48,45,46);(2,3)(4,11)(8,10)(12,13)(17,19,23,21)(18,24,22,26)(20,29,30,27)(25,32,28,31)(33,35)(34,38)(36,43)(37,45)(39,41)(40,46)(42,47)(44,48);(2,3)(7,9)(8,10)(15,16)(34,36)(35,41)(40,44)(42,45);(1,3,6,2)(4,7,11,9)(5,10,14,8)(12,15,13,16)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,36,39,34)(35,38,41,43)(37,40,47,44)(42,46,45,48);(1,4)(2,7)(3,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,21,23,19)(18,24,22,26)(20,27,30,29)(25,32,28,31)(33,37,39,47)(34,40,36,44)(35,42,41,45)(38,48,43,46);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)(33,38)(34,35)(36,41)(37,46)(39,43)(40,45)(42,44)(47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=gi,e^2=gh,f^2=g,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, 10154113);