#10320350

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

4382 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ghi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

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, 10227546);