#8096265

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

4559 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:

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

Polycyclic presentation:

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

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, 8003461);