#10156219

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

4411 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)(11,16)(12,14)(13,18)(15,17)(19,21)(20,23)(22,24)(25,26)(27,29)(28,31)(30,36)(32,34)(33,38)(35,39)(37,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,12)(10,16,17,14)(13,20,22,19)(18,23,24,21)(25,27)(26,32)(28,29)(30,35)(31,34)(33,39)(36,37)(38,40);(1,3,5,4)(2,6,7,8)(10,17)(11,12)(18,24)(19,20)(26,28)(27,34)(33,36)(35,40);(1,2)(3,6)(4,8)(5,7)(9,12,15,11)(10,14,17,16)(13,19,22,20)(18,21,24,23)(25,28,31,26)(27,29,34,32)(30,36,38,33)(35,37,40,39);(1,4,5,3)(2,6,7,8)(9,13,15,22)(10,18,17,24)(11,19,12,20)(14,23,16,21)(25,29)(26,27)(28,34)(30,37)(31,32)(33,35)(36,40)(38,39);(1,2)(3,6)(4,8)(5,7)(9,14)(10,11)(12,17)(13,21)(15,16)(18,20)(19,24)(22,23)(25,30,31,38)(26,33,28,36)(27,35,34,40)(29,37,32,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10063415);