#8741650

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

4580 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8648846);