#8108131

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

5016 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)(6,8)(10,13)(14,16)(17,18)(19,24)(20,29)(21,30)(22,23)(25,27)(26,31)(28,32);(2,5)(6,8)(11,15)(14,16)(18,23)(24,32)(25,29)(26,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,24,23,32)(20,21,27,31)(25,26,29,30);(9,11,12,15)(10,14,13,16)(17,20,22,27)(18,25,23,29)(19,21,28,31)(24,26,32,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31);(1,3)(2,6)(4,7)(5,8)(17,21)(18,26)(19,27)(20,28)(22,31)(23,30)(24,29)(25,32)

Polycyclic presentation:

2^9|c^2=g,d^2=gh,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, 8015327);