#9781258

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

4813 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,f^2=gh,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, 9688454);