#8757525

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

4357 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gi,e^2=gi,f^2=gh,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, 8664721);