#8180408

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

4137 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087604);