#10042532

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

4397 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:

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

Polycyclic presentation:

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

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, 9949728);