#9237031

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

4333 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=g,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, 9144227);