#8828869

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

4779 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8736065);