#8094597

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

4329 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 8001793);