#10156218

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	#54757	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4196 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10063414);