#8188088

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

4406 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=ghi,e^2=h,f^2=gh,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, 8095284);