#9028802

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

4652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gi,d^2=gh,e^2=gh,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, 8935998);