#10504471

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

4507 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10411667);