#8739428

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

4306 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=h,e^2=h,f^2=gh,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, 8646624);