#10328210

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

5107 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10235406);