#10329416

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
prime radical	#51410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4293 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=g,e^2=gh,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, 10236612);