#8831227

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

4941 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:

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

Polycyclic presentation:

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

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, 8738423);