#8196401

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4262 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)(19,21)(25,26,30,33)(27,28,31,35)(29,32,38,40)(34,36,37,39);(1,2,6,5)(3,8,4,7)(9,10,14,12)(11,15,17,18)(13,16,22,20)(19,23,21,24)(27,31)(28,35)(34,37)(36,39);(1,3,6,4)(2,7,5,8)(9,11)(10,15)(12,18)(13,21)(14,17)(16,24)(19,22)(20,23)(25,27,30,31)(26,28,33,35)(29,37,38,34)(32,39,40,36);(9,10,14,12)(11,15,17,18)(13,16,22,20)(19,23,21,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,37)(34,40)(38,39);(1,4,6,3)(2,7,5,8)(9,12,14,10)(11,18,17,15)(13,20,22,16)(19,24,21,23);(1,5,6,2)(3,7,4,8)(9,13,14,22)(10,16,12,20)(11,19,17,21)(15,23,18,24)(25,29,30,38)(26,32,33,40)(27,34,31,37)(28,36,35,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=g,e^2=h,f^2=gi,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, 8103597);