#8425569

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

4501 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,e^2=h,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, 8332765);