#8108009

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4278 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)(3,6)(4,8)(5,7)(9,10)(11,16)(12,14)(13,20)(15,19)(17,21)(18,22)(23,24)(25,26,30,33)(27,31,35,28)(29,37,38,32)(34,36,40,39);(2,3)(7,8)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(2,3)(7,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(1,3,6,2)(4,8,5,7)(9,12,14,10)(11,16,19,15)(13,20,21,17)(18,23,24,22)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(1,4,6,5)(2,7,3,8);(1,5,6,4)(2,8,3,7)(9,13,14,21)(10,17,12,20)(11,18,19,24)(15,22,16,23)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39)

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=h,f^2=gi,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, 8015205);