#8828706

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

4959 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8735902);