#8108124

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

4614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 8015320);