#8665304

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

4315 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8572500);