#10480978

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

4088 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=i,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, 10388174);