#8904513

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

5097 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,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, 8811709);