#8446051

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

4429 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,e^2=hi,f^2=g,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, 8353247);