#10247341

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

4499 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)(3,8)(4,6)(5,7)(9,10,15,18)(11,12,13,16)(14,22,23,17)(19,24,21,20)(25,26,31,29)(27,32,33,28)(30,37,38,34)(35,36,40,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,18)(13,15)(14,19)(17,24)(20,22)(21,23)(25,27)(26,32)(28,29)(30,35)(31,33)(34,39)(36,37)(38,40);(9,12,15,16)(10,11,18,13)(14,20,23,24)(17,19,22,21)(25,28,31,32)(26,33,29,27)(30,36,38,39)(34,40,37,35);(1,2,5,7)(3,8,4,6)(9,12)(10,11)(13,18)(14,20)(15,16)(17,19)(21,22)(23,24)(25,28)(26,27)(29,33)(30,36)(31,32)(34,35)(37,40)(38,39);(1,2)(3,6)(4,8)(5,7)(9,13,15,11)(10,16,18,12)(14,19,23,21)(17,20,22,24)(25,29,31,26)(27,32,33,28)(30,34,38,37)(35,36,40,39);(1,4,5,3)(2,6,7,8)(9,14,15,23)(10,17,18,22)(11,19,13,21)(12,20,16,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=ghi,e^2=h,f^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, 10154537);