#8733941

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

4625 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^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, 8641137);