#8108738

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

4890 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=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, 8015934);