#8862392

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

4680 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8769588);