#8745937

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

4310 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=ghi,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, 8653133);