#10504481

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

4172 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)(3,12)(4,10)(5,16)(6,14)(7,11)(8,15)(9,13)(17,18)(19,20)(21,25)(22,23)(24,27)(26,28)(29,30)(31,32)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,3)(2,6)(4,5)(7,14)(8,16)(9,10)(11,12)(13,15)(18,23)(20,24)(25,32)(28,29)(33,34,39,36)(35,40,38,42)(37,41,46,44)(43,47,45,48);(1,4)(2,8)(3,13)(5,14)(6,9)(7,15)(10,11)(12,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,29)(28,32)(30,31)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,5,7,13)(2,9,11,16)(3,4,14,15)(6,10,12,8)(17,20,22,24)(18,19,23,27)(21,29,31,28)(25,30,32,26)(33,36,39,34)(35,42,38,40)(37,41,46,44)(43,47,45,48);(1,4)(2,10)(3,5)(6,9)(7,15)(8,11)(12,16)(13,14)(17,21,22,31)(18,25,23,32)(19,26,27,30)(20,28,24,29)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,6)(2,3)(4,9)(5,10)(7,12)(8,13)(11,14)(15,16)(17,19)(18,24)(20,23)(21,30)(22,27)(25,28)(26,31)(29,32)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,e^2=hi,f^2=gh,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, 10411677);