#9717747

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

5190 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9624943);