#8139664

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

4884 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)(9,10,12,13)(11,15,16,14)(17,18,24,27)(19,28,31,25)(20,22,21,32)(23,30,29,26);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15)(17,19,24,31)(18,25,27,28)(20,29,21,23)(22,30,32,26);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,24,21)(18,22,27,32)(19,29,31,23)(25,30,28,26);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,21,24,20)(18,22,27,32)(19,23,31,29)(25,30,28,26);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,22)(18,21)(19,30)(20,27)(23,25)(24,32)(26,31)(28,29);(1,2)(3,5)(4,6)(7,8)(17,23)(18,26)(19,20)(21,31)(22,25)(24,29)(27,30)(28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046860);