#10412325

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

4344 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:

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

Polycyclic presentation:

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

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, 10319521);