#8634810

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

4827 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8542006);