#10041414

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

4800 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)(3,7)(4,6)(5,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,30,31)(27,29,33,38)(28,32,35,39)(34,37,40,36);(1,2,4,6)(3,5,8,7)(12,19)(15,17)(20,24)(22,23)(26,31)(27,33)(28,35)(36,37);(1,2)(3,5)(4,6)(7,8)(10,11)(15,17)(16,18)(22,23)(26,31)(27,33)(28,35)(36,37);(1,3)(2,5)(4,8)(6,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27)(26,29)(28,34)(30,33)(31,38)(32,37)(35,40)(36,39);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,40)(36,38);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,27)(28,36)(30,38)(31,33)(32,40)(34,39)(35,37)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=ghi,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, 9948610);