#10480900

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

4117 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gi,e^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, 10388096);