#8460748

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

4624 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,e^2=gi,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, 8367944);