#8908420

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

4545 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,13)(14,22)(15,17)(18,23)(19,21)(20,24)(25,26)(27,33)(28,37)(29,34)(30,35)(31,36)(32,38)(39,40);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,16,17,13)(14,19,23,20)(18,24,22,21)(25,27)(26,31)(28,34)(29,32)(30,36)(33,35)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(10,17)(11,12)(14,23)(21,24)(28,38)(29,40)(32,37)(34,39);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,13,17,16)(14,20,23,19)(18,21,22,24)(25,28)(26,32)(27,34)(29,31)(30,38)(33,40)(35,37)(36,39);(9,13)(10,12)(11,17)(14,21)(15,16)(18,20)(19,22)(23,24)(25,27)(26,33)(28,39)(29,32)(30,36)(31,35)(34,38)(37,40);(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,24)(17,22)(25,29)(26,34)(27,32)(28,31)(30,40)(33,38)(35,39)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8815616);