#8831198

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

4567 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=g,f^2=h,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, 8738394);