#10045518

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

4505 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)(3,5)(4,7)(6,8)(9,10,14,16)(11,15,18,24)(12,21,17,13)(19,20,22,23)(25,26,31,32)(27,34,30,29)(28,33,37,39)(35,40,36,38);(1,3)(2,4)(5,8)(6,7)(9,11,14,18)(10,15,16,24)(12,22,17,19)(13,20,21,23)(25,27,31,30)(26,29,32,34)(28,36,37,35)(33,40,39,38);(1,4,6,5)(2,7,8,3)(10,16)(11,18)(12,17)(20,23)(26,32)(28,37)(29,34)(35,36);(1,2)(3,5)(4,7)(6,8)(9,12,14,17)(10,13,16,21)(11,19,18,22)(15,23,24,20)(25,28)(26,33)(27,35)(29,38)(30,36)(31,37)(32,39)(34,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,15)(12,19)(13,23)(14,18)(16,24)(17,22)(20,21)(25,29)(26,27)(28,38)(30,32)(31,34)(33,35)(36,39)(37,40);(1,5,6,4)(2,3,8,7)(9,13)(10,17)(11,20)(12,16)(14,21)(15,19)(18,23)(22,24)(25,30,31,27)(26,34,32,29)(28,36,37,35)(33,40,39,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=i,f^2=g,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, 9952714);