#9040760

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

4539 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947956);