#8152217

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

4922 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8059413);