#9062198

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

4831 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969394);