#8757521

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

4533 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,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, 8664717);