#10482501

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

4234 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10389697);