#10352609

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

4349 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=hi,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, 10259805);