#8003815

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

4717 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)(9,10,13,12)(11,15,16,14)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32);(1,3,5,8)(2,6,4,7)(9,11,13,16)(10,14,12,15)(18,23)(20,24)(26,29)(28,31);(1,3)(2,7)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(18,23)(19,25)(26,29)(27,32);(1,3,5,8)(2,6,4,7)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28);(1,3)(2,6)(4,7)(5,8)(9,10,13,12)(11,14,16,15)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,15,16,14)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=h,e^2=g,f^2=gh,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, 7911011);