#8147845

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

4676 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)(4,6)(5,8)(14,22)(17,21)(19,20)(23,24)(26,29)(27,28)(34,37)(35,36);(1,3)(2,5)(4,8)(6,7)(9,10,15,13)(11,18,12,16)(14,17,22,21)(19,24,20,23)(25,26,31,29)(27,33,28,32)(30,34,38,37)(35,40,36,39);(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(2,6)(5,7)(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24)(25,28,31,27)(26,33,29,32)(30,36,38,35)(34,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,13,15,10)(11,18,12,16)(14,21,22,17)(19,24,20,23)(25,29,31,26)(27,33,28,32)(30,37,38,34)(35,40,36,39);(1,3)(2,5)(4,8)(6,7)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30,31,38)(26,34,29,37)(27,35,28,36)(32,39,33,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8055041);