#8898155

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

5896 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)(3,4)(5,6)(7,8)(17,18,23,25)(19,24,27,29)(20,22,30,31)(21,26,32,28);(2,6)(3,7)(9,10)(11,13)(12,14)(15,16)(17,19,23,27)(18,24,25,29)(20,28,30,26)(21,31,32,22);(2,6)(4,8)(9,10,12,14)(11,13,15,16)(18,25)(19,27)(22,31)(26,28);(2,6)(3,7)(11,15)(13,16)(17,20)(18,22)(19,28)(21,24)(23,30)(25,31)(26,27)(29,32);(1,3,5,7)(2,4,6,8)(9,11)(10,13)(12,15)(14,16)(17,21)(18,26)(19,22)(20,29)(23,32)(24,30)(25,28)(27,31);(1,4)(2,7)(3,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,22)(18,20)(19,21)(23,31)(24,28)(25,30)(26,29)(27,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805351);