#8406580

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

4850 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8313776);