#8458721

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

4540 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)(3,4)(5,6)(7,8)(12,20)(16,23)(19,24)(21,22)(26,30)(31,37)(32,38)(39,40);(2,6)(3,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(26,30)(27,33)(32,38)(34,35);(1,3,5,7)(2,4,6,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27)(26,31)(28,35)(29,33)(30,37)(32,40)(34,36)(38,39);(9,12,14,20)(10,16,18,23)(11,19,17,24)(13,21,15,22);(9,13,14,15)(10,17,18,11)(12,21,20,22)(16,24,23,19)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40)

Polycyclic presentation:

2^9|c^2=g,e^2=ghi,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, 8365917);