#8195912

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

4421 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8103108);