#10218579

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

4728 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10125775);