#8422146

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

5026 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8329342);