#10158234

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

4654 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=gh,e^2=ghi,f^2=gh,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, 10065430);