#8105998

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
prime radical	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4584 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013194);