#8990721

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

5086 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)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31);(3,7)(6,8)(9,10)(11,13)(12,14)(15,16)(18,20)(23,25)(24,28)(30,31);(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(18,20)(19,26)(21,29)(30,31);(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=g,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, 8897917);