#10218495

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

4354 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10125691);