#8840321

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

4402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8747517);