#8891276

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

4847 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=g,e^2=hi,f^2=ghi,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, 8798472);