#8115328

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

4369 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022524);