#8108324

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

4996 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015520);