#9368157

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

4986 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)(3,7)(4,8)(5,6)(9,10)(11,12)(13,14)(15,16)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32);(2,6)(4,7)(10,14)(11,15)(17,19,23,21)(18,24,20,25)(22,29,30,27)(26,32,28,31);(2,6)(3,8)(9,11,13,15)(10,12,14,16)(19,21)(24,25)(27,29)(31,32);(1,3)(2,4)(5,8)(6,7)(9,12)(10,15)(11,14)(13,16)(17,20,23,18)(19,25,21,24)(22,28,30,26)(27,32,29,31);(1,2)(3,4)(5,6)(7,8)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31);(1,4)(2,3)(5,7)(6,8)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9275353);