#10480727

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

4775 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=i,e^2=g,f^2=g,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, 10387923);