#10089178

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

4894 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,f^2=h,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, 9996374);