#8437808

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

4421 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)(3,7)(4,5)(6,8)(9,10,16,15)(11,17,14,19)(12,22,13,18)(20,24,21,23)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(2,5)(6,7)(9,11,16,14)(10,17,15,19)(12,21,13,20)(18,24,22,23)(26,31)(27,34)(33,36)(35,40);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,15,22)(11,20,14,21)(17,23,19,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,13,16,12)(10,18,15,22)(11,21,14,20)(17,23,19,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(9,14,16,11)(10,19,15,17)(12,21,13,20)(18,24,22,23)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,15,16,10)(11,19,14,17)(12,22,13,18)(20,24,21,23)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=h,e^2=h,f^2=h,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, 8345004);