#8924726

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

4466 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)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(25,26,30,33)(27,34,29,31)(28,36,38,32)(35,39,37,40);(1,3,5,8)(2,6,7,4)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,38)(37,40);(1,2)(3,4)(5,7)(6,8)(10,14)(12,21)(15,17)(18,24)(25,27,30,29)(26,31,33,34)(28,37,38,35)(32,40,36,39);(1,4)(2,3)(5,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,28,30,38)(26,32,33,36)(27,35,29,37)(31,39,34,40);(1,3)(2,4)(5,8)(6,7)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,29,30,27)(26,31,33,34)(28,35,38,37)(32,40,36,39);(1,4)(2,3)(5,6)(7,8)(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,38)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831922);