#8096966

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

4490 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:

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

Polycyclic presentation:

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

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, 8004162);