#9105307

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

4904 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

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, 9012503);