#10143814

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

4905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=g,d^2=g,e^2=i,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, 10051010);