#10463944

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

4725 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)(11,16)(12,17)(13,18)(14,15)(19,21)(20,22)(23,24)(25,26,30,34)(27,35,38,31)(28,39,37,32)(29,33,36,40);(10,15)(11,19)(17,20)(18,24)(25,27,30,38)(26,31,34,35)(28,29,37,36)(32,33,39,40);(1,2)(3,6)(4,5)(7,8)(25,28)(26,32)(27,36)(29,38)(30,37)(31,40)(33,35)(34,39);(1,3,4,7)(2,6,5,8)(9,11,14,19)(10,16,15,21)(12,18,22,24)(13,20,23,17)(25,28)(26,32)(27,29)(30,37)(31,33)(34,39)(35,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,18)(13,16)(14,22)(15,20)(19,24)(21,23)(25,29)(26,33)(27,37)(28,38)(30,36)(31,39)(32,35)(34,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,20)(12,21)(14,23)(15,24)(16,22)(17,19)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gh,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, 10371140);