#8741901

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

4581 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)(3,4)(5,6)(7,8)(9,10,15,17)(11,18,13,16)(12,14,19,24)(20,21,22,23)(25,26,30,34)(27,29,33,31)(28,32,36,39)(35,38,40,37);(2,6)(3,7)(9,11,15,13)(10,16,17,18)(12,22,19,20)(14,21,24,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,37)(36,40)(38,39);(1,3,5,7)(2,4,6,8)(10,17)(12,19)(16,18)(20,22)(28,36)(32,39)(35,40)(37,38);(1,4)(2,7)(3,6)(5,8)(9,12)(10,14)(11,20)(13,22)(15,19)(16,23)(17,24)(18,21)(25,28)(26,32)(27,35)(29,38)(30,36)(31,37)(33,40)(34,39);(9,13,15,11)(10,18,17,16)(12,22,19,20)(14,21,24,23)(25,27)(26,29)(28,35)(30,33)(31,34)(32,38)(36,40)(37,39);(9,14)(10,19)(11,21)(12,17)(13,23)(15,24)(16,20)(18,22)(25,29)(26,33)(27,34)(28,37)(30,31)(32,35)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,e^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, 8649097);