#8461227

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

4600 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,e^2=hi,f^2=i,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, 8368423);