#10368412

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

4321 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10275608);