#8429712

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

4642 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336908);