#10521352

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

4520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=ghi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10428548);