#9271789

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

4531 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:

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

Polycyclic presentation:

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

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, 9178985);