#10312446

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

4378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=g,f^2=g,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, 10219642);