#10446491

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

4734 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:

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

Polycyclic presentation:

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

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, 10353687);