#8460716

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

4479 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=g,e^2=hi,f^2=hi,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, 8367912);