#10219586

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

4368 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10126782);