#8942372

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

4619 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=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, 8849568);