#8358677

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

4392 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8265873);