#10143809

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

4954 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)(17,18,23,28)(19,27,22,24)(20,30,31,25)(21,26,32,29);(2,6)(3,7)(9,10)(11,12)(13,15)(14,16)(17,19,23,22)(18,24,28,27)(20,29,31,26)(21,30,32,25);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,23,31)(18,25,28,30)(19,29,22,26)(21,27,32,24);(2,6)(3,7)(9,11)(10,12)(13,16)(14,15)(17,20)(18,25)(19,29)(21,24)(22,26)(23,31)(27,32)(28,30);(1,3,5,7)(2,4,6,8)(9,12,13,14)(10,11,15,16)(17,21)(18,26)(19,25)(20,27)(22,30)(23,32)(24,31)(28,29);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,22,23,19)(18,27,28,24)(20,26,31,29)(21,25,32,30)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^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, 10051005);