#8505673

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

4711 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412869);