#10482488

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

4078 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=gi,e^2=g,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, 10389684);