#10143819

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

5045 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)(17,18,22,23)(19,20,25,24)(21,29,31,26)(27,30,32,28);(2,6)(3,7)(9,10)(11,12)(13,15)(14,16)(18,23)(20,24)(26,29)(28,30);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(18,23)(19,25)(26,29)(27,32);(2,6)(3,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,24)(20,23)(21,27)(22,25)(26,30)(28,29)(31,32);(1,3,5,7)(2,4,6,8)(9,12,13,14)(10,11,15,16)(17,20,22,24)(18,25,23,19)(21,30,31,28)(26,27,29,32);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,21,22,31)(18,26,23,29)(19,27,25,32)(20,28,24,30)

Polycyclic presentation:

2^9|a^2=g,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, 10051015);