#8492921

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

4306 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=h,e^2=i,f^2=g,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, 8400117);