#8664987

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

5065 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8572183);