#10142142

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

4515 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=ghi,e^2=h,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, 10049338);