#8691723

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

4659 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8598919);