#10068927

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

4886 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9976123);