#8464231

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

4525 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=gh,e^2=i,f^2=gi,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, 8371427);