#9039288

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

4930 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8946484);