#8757898

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

4829 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)(3,5,7,8)(17,18)(19,21)(20,25)(22,23)(24,26)(27,29)(28,30)(31,32);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,13)(14,16)(18,23)(20,28)(21,24)(27,32);(1,2)(3,5)(4,6)(7,8)(10,13)(14,15)(18,23)(19,26)(20,28)(29,31);(9,10)(11,14)(12,13)(15,16)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,31,30,29);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,28)(18,25,23,30)(19,27,26,32)(21,29,24,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,21,22,24)(18,26,23,19)(20,29,28,31)(25,32,30,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8665094);