#8926205

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

4403 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833401);