#8904627

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

5108 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)(11,15)(12,16)(13,14)(17,18,24,23)(19,27,21,25)(20,26,29,32)(22,28,30,31);(1,2)(3,6)(4,7)(5,8)(10,14)(11,16)(17,19,24,21)(18,25,23,27)(20,22,29,30)(26,31,32,28);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,20,24,29)(18,26,23,32)(19,22,21,30)(25,31,27,28);(3,8)(5,6)(9,12)(10,11)(13,15)(14,16)(17,21,24,19)(18,27,23,25)(20,22,29,30)(26,31,32,28);(1,3)(2,6)(4,8)(5,7)(9,10)(11,12)(13,14)(15,16)(17,22)(18,28)(19,29)(20,21)(23,31)(24,30)(25,26)(27,32);(1,2)(3,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,23,24,18)(19,27,21,25)(20,26,29,32)(22,31,30,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8811823);