#10328742

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

4244 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=g,e^2=g,f^2=g,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, 10235938);