#9057777

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

5194 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8964973);