#9027899

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

4643 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=i,d^2=g,e^2=g,f^2=i,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, 8935095);