#9779477

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
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4543 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9686673);