#8388297

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

4535 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=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, 8295493);