#8734297

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

4721 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,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, 8641493);