#9267056

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

4805 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)(11,16)(12,18)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26)(27,29)(28,31)(30,36)(32,34)(33,38)(35,39)(37,40);(1,2,5,6)(3,4,8,7)(9,11,15,12)(10,16,17,18)(13,22,19,20)(14,23,24,21)(25,27)(26,32)(28,29)(30,35)(31,34)(33,39)(36,37)(38,40);(1,2,5,6)(3,7,8,4)(10,17)(11,12)(13,19)(21,23)(26,28)(27,34)(33,36)(35,40);(1,3)(2,4)(5,8)(6,7)(9,12,15,11)(10,18,17,16)(13,22,19,20)(14,23,24,21)(25,28,31,26)(27,29,34,32)(30,36,38,33)(35,37,40,39);(1,2)(3,4)(5,6)(7,8)(9,13,15,19)(10,14,17,24)(11,20,12,22)(16,21,18,23)(25,29)(26,27)(28,34)(30,37)(31,32)(33,35)(36,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,14)(10,19)(11,21)(12,23)(13,17)(15,24)(16,22)(18,20)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=i,e^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, 9174252);