#8165186

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

4458 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)(3,7)(4,8)(5,6)(9,10)(11,17)(12,13)(14,22)(15,16)(18,23)(19,20)(21,24)(25,26,29,33)(27,31,34,30)(28,36,37,32)(35,38,40,39);(2,6)(3,8)(9,11,15,13)(10,12,16,17)(14,19,23,21)(18,20,22,24)(27,34)(30,31)(35,40)(38,39);(1,3)(2,4)(5,8)(6,7)(10,16)(12,17)(18,22)(20,24)(25,27,29,34)(26,30,33,31)(28,35,37,40)(32,38,36,39);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,16)(14,20)(15,17)(18,19)(21,22)(23,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(1,4)(2,3)(5,7)(6,8)(9,13,15,11)(10,17,16,12)(14,21,23,19)(18,24,22,20)(25,26)(27,30)(28,32)(29,33)(31,34)(35,38)(36,37)(39,40);(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,22)(17,24)(25,28,29,37)(26,32,33,36)(27,35,34,40)(30,38,31,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8072382);