#8657246

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

5476 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8564442);