#8464194

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

4515 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,22)(13,17)(14,18)(16,19)(20,24)(21,23)(26,31)(28,32)(34,37)(36,39);(2,5)(6,7)(9,11,16,14)(10,15,19,18)(12,21,13,20)(17,24,22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,17,19,22)(11,20,14,21)(15,23,18,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(1,3)(2,6)(4,8)(5,7)(9,13,16,12)(10,17,19,22)(11,21,14,20)(15,23,18,24)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(9,14,16,11)(10,18,19,15)(12,21,13,20)(17,24,22,23)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(9,15,16,18)(10,11,19,14)(12,23,13,24)(17,20,22,21)(25,29,30,38)(26,34,31,37)(27,35,33,40)(28,36,32,39)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=h,e^2=gi,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, 8371390);