#8188085

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

4429 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)(10,12)(11,17)(16,20)(19,23)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(1,2,6,5)(3,8,4,7)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,3,6,4)(2,7,5,8)(9,10,14,12)(11,15,17,18)(13,20,21,16)(19,24,23,22)(25,27,31,29)(26,33,28,32)(30,35,38,37)(34,40,36,39);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(1,4,6,3)(2,7,5,8)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,29,31,27)(26,32,28,33)(30,37,38,35)(34,39,36,40);(1,5,6,2)(3,7,4,8)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,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, 8095281);