#8436952

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

4363 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,e^2=h,f^2=g,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, 8344148);