#8618167

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

5283 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8525363);