#8891805

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

4933 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,e^2=h,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, 8799001);