#8341830

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

4338 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8249026);