#10218347

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

5882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=gh,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, 10125543);