#9379989

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

4816 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)(3,4)(5,6)(7,8)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26,30,33)(27,31,34,28)(29,32,38,40)(35,39,37,36);(2,6)(3,7)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(25,27,30,34)(26,31,33,28)(29,37,38,35)(32,36,40,39);(1,3,5,7)(2,4,6,8)(10,15)(11,18)(16,23)(19,20)(27,34)(28,31)(35,37)(36,39);(1,4)(2,7)(3,6)(5,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,28)(26,27)(29,36)(30,31)(32,35)(33,34)(37,40)(38,39);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,37)(38,40);(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,40)(34,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,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=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9287185);