#9028249

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

5031 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)(11,14)(12,13)(15,16)(17,18,22,26)(19,21,25,23)(20,24,28,32)(27,30,29,31);(1,2,5,4)(3,7,8,6)(9,11,13,15)(10,14,12,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30);(1,3,5,8)(2,6,4,7)(10,12)(11,15)(19,25)(20,28)(21,23)(24,32);(3,8)(6,7)(9,11,13,15)(10,14,12,16)(17,19,22,25)(18,23,26,21)(20,29,28,27)(24,30,32,31);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,28)(18,24,26,32)(19,27,25,29)(21,30,23,31);(1,4,5,2)(3,6,8,7)(9,12,13,10)(11,16,15,14)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,29,32,27)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8935445);