#8985284

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

4861 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)(3,7)(4,5)(6,8)(18,23)(19,25)(26,30)(27,32);(3,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30);(1,3,5,8)(2,6,4,7)(10,13)(11,15)(19,25)(20,24)(21,29)(26,30);(3,8)(6,7)(9,11,12,15)(10,14,13,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27);(1,4,5,2)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=g,f^2=h,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, 8892480);