#10196247

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

4053 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:

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

Polycyclic presentation:

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

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, 10103443);