#8189402

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

4403 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,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, 8096598);