#9127634

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

5074 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9034830);