#8505425

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

4381 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)(3,8)(4,7)(5,6)(9,10)(11,19)(12,13)(14,16)(15,18)(17,23)(20,22)(21,24)(25,26,31,28)(27,32,37,35)(29,38,40,33)(30,34,36,39);(2,6)(3,7)(9,11,15,14)(10,16,18,19)(12,20,23,21)(13,24,17,22)(25,27)(26,32)(28,35)(29,30)(31,37)(33,39)(34,38)(36,40);(9,12)(10,17)(11,20)(13,18)(14,21)(15,23)(16,22)(19,24)(25,26,31,28)(27,35,37,32)(29,33,40,38)(30,34,36,39);(2,6)(4,8)(9,12)(10,17)(11,21)(13,18)(14,20)(15,23)(16,24)(19,22)(25,28,31,26)(27,35,37,32)(29,38,40,33)(30,34,36,39);(1,3,5,7)(2,4,6,8)(9,13,15,17)(10,12,18,23)(11,22,14,24)(16,21,19,20)(25,29)(26,33)(27,36)(28,38)(30,37)(31,40)(32,34)(35,39);(1,4)(2,7)(3,6)(5,8)(9,14,15,11)(10,16,18,19)(12,21,23,20)(13,24,17,22)(25,30,31,36)(26,34,28,39)(27,29,37,40)(32,38,35,33)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=i,e^2=g,f^2=hi,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, 8412621);