#8421776

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

4517 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,f^2=g,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, 8328972);