#10121166

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

5052 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=hi,e^2=g,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, 10028362);