#8437787

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

4384 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=g,e^2=h,f^2=h,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, 8344983);