#10218519

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

4189 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10125715);