#8422177

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

5082 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,e^2=h,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, 8329373);