#10247215

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

4264 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154411);