#10250333

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

5132 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)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28);(1,3,6,4)(2,7,8,5)(9,10)(11,16)(12,14)(13,15)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28);(9,11,13,14)(10,12,15,16)(18,25)(19,30)(20,27)(28,32);(2,8)(3,4)(9,10,13,15)(11,12,14,16)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32);(1,4,6,3)(2,7,8,5)(9,10)(11,12)(13,15)(14,16)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29);(1,5,6,7)(2,4,8,3)(9,12)(10,14)(11,15)(13,16)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=i,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, 10157529);