#9041015

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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4590 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948211);