#10320379

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

4445 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)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26,31,29)(27,28,32,36)(30,33,39,37)(34,35,40,38);(1,3)(2,7)(4,5)(6,8)(11,17)(14,21)(18,23)(22,24)(25,27,31,32)(26,28,29,36)(30,34,39,40)(33,35,37,38);(1,2,6,4)(3,7,8,5)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,28)(26,27)(29,32)(30,35)(31,36)(33,34)(37,40)(38,39);(2,4)(3,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,29,31,26)(27,28,32,36)(30,37,39,33)(34,35,40,38);(1,4,6,2)(3,7,8,5)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,28)(26,32)(27,29)(30,38)(31,36)(33,34)(35,39)(37,40);(1,5,6,7)(2,3,4,8)(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,30,31,39)(26,33,29,37)(27,34,32,40)(28,35,36,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=ghi,d^2=i,e^2=g,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, 10227575);