#8990720

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

5168 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8897916);