#8395292

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

4865 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)(3,6)(4,5)(7,8)(17,18)(19,20)(21,23)(22,25)(24,26)(27,28)(29,32)(30,31);(1,2,4,5)(3,6,7,8)(9,10)(11,16)(12,15)(13,14)(17,19,24,25)(18,20,26,22)(21,27,29,31)(23,28,32,30);(3,7)(6,8)(10,14)(12,16)(17,20,24,22)(18,25,26,19)(21,28,29,30)(23,31,32,27);(2,5)(6,8)(10,14)(11,15)(17,21)(18,23)(19,27)(20,28)(22,30)(24,29)(25,31)(26,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,22,24,20)(18,19,26,25)(21,28,29,30)(23,31,32,27);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,14)(13,16)(17,23)(18,21)(19,28)(20,27)(22,31)(24,32)(25,30)(26,29)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8302488);