#8429496

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

4501 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)(3,4,7,8)(9,10,16,19)(11,17,14,23)(12,15,13,18)(20,24,21,22)(25,26,31,28)(27,29,35,32)(30,33,39,36)(34,37,40,38);(2,6)(3,7)(9,11,16,14)(10,17,19,23)(12,21,13,20)(15,22,18,24)(25,27,31,35)(26,32,28,29)(30,34,39,40)(33,38,36,37);(1,3)(2,4)(5,7)(6,8)(9,12,16,13)(10,18,19,15)(11,20,14,21)(17,22,23,24)(25,28,31,26)(27,29,35,32)(30,36,39,33)(34,37,40,38);(1,4)(2,3)(5,8)(6,7)(9,13,16,12)(10,18,19,15)(11,21,14,20)(17,22,23,24)(25,29)(26,27)(28,35)(30,38)(31,32)(33,40)(34,36)(37,39);(9,14,16,11)(10,17,19,23)(12,21,13,20)(15,24,18,22);(9,15,16,18)(10,13,19,12)(11,22,14,24)(17,20,23,21)(25,30,31,39)(26,33,28,36)(27,34,35,40)(29,37,32,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8336692);