#10504518

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

4822 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)(9,10,13,16)(11,17,18,14)(12,15,22,21)(19,24,20,23)(25,26,30,33)(27,34,28,31)(29,37,38,32)(35,39,36,40);(11,18)(14,17)(19,20)(23,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,17)(20,22)(21,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(3,7)(6,8)(9,11)(10,14)(12,20)(13,18)(15,24)(16,17)(19,22)(21,23)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(1,3)(2,6)(4,7)(5,8)(9,12,13,22)(10,15,16,21)(11,19,18,20)(14,23,17,24)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,21)(13,16)(15,22)(17,18)(19,24)(20,23)(25,29,30,38)(26,32,33,37)(27,35,28,36)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=i,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, 10411714);