#8922569

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

4567 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8829765);