#9123903

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

5090 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,e^2=gh,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, 9031099);