#8186263

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

5007 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=gi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8093459);