#8016655

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

5167 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=g,e^2=gi,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

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, 7923851);