#8003817

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

4531 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)(9,10,13,16)(11,14,12,15)(17,18)(19,31)(20,26)(21,27)(22,28)(23,29)(24,30)(25,32);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,25)(20,23)(21,28)(22,26)(24,32)(27,29)(30,31);(1,3)(2,7)(4,6)(5,8)(17,20,24,21)(18,26,30,27)(19,28,32,23)(22,25,29,31);(1,3,5,8)(2,6,4,7)(9,11,13,12)(10,14,16,15)(17,21,24,20)(18,27,30,26)(19,28,32,23)(22,25,29,31);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,16,15)(17,22,24,29)(18,28,30,23)(19,26,32,27)(20,25,21,31);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,16,14)(17,23,24,28)(18,29,30,22)(19,20,32,21)(25,27,31,26)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=gh,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, 7911013);