#9720453

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

4684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,e^2=ghi,f^2=gi,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, 9627649);