#9245145

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

4263 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)(3,7)(4,8)(5,6)(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,40)(37,39);(2,6)(4,7)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(26,31)(32,35)(33,34)(39,40);(2,6)(3,8)(10,15)(11,18)(16,21)(19,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,27,30,38)(26,32,31,35)(28,36,29,37)(33,39,34,40);(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,28,30,29)(26,33,31,34)(27,36,38,37)(32,39,35,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,29,30,28)(26,34,31,33)(27,37,38,36)(32,40,35,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,d^2=gh,e^2=gh,f^2=gh,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, 9152341);