#8417144

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

4234 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8324340);