#8678944

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

5074 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,d^2=h,f^2=gi,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, 8586140);