#10320412

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

4271 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,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, 10227608);