#8527423

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

5119 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:

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

Polycyclic presentation:

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

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, 8434619);