#9715474

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

4779 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:

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

Polycyclic presentation:

2^9|a^2=gh,e^2=gi,f^2=h,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, 9622670);