#10480814

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

4174 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)(3,8)(4,9)(5,10)(6,7)(11,12)(13,15)(14,16)(18,23)(19,25)(21,29)(28,31)(36,44)(39,45)(42,48)(46,47);(2,7)(3,10)(9,15)(11,16)(17,18,22,23)(19,20,25,24)(21,26,29,30)(27,28,32,31)(33,34)(35,40)(36,45)(37,41)(38,43)(39,44)(42,46)(47,48);(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,48)(45,47);(1,3,6,10)(2,8,7,5)(4,11,13,16)(9,12,15,14)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,32)(30,31)(33,36,37,44)(34,39,41,45)(35,42,43,48)(38,46,40,47);(1,4)(2,9)(3,11)(5,14)(6,13)(7,15)(8,12)(10,16)(17,20)(18,25)(19,23)(21,28)(22,24)(26,32)(27,30)(29,31)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,5,6,8)(2,10,7,3)(4,12,13,14)(9,11,15,16)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,34)(35,38)(36,39)(37,41)(40,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|b^2=h,d^2=gi,f^2=g,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, 10388010);