#10322237

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

4367 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10)(11,19)(12,16)(13,14)(15,18)(17,24)(20,23)(21,22)(26,32)(27,28)(30,34)(37,39);(1,2)(3,7)(4,5)(6,8)(25,26)(27,36)(28,33)(29,34)(30,35)(31,32)(37,38)(39,40);(2,5)(6,7)(9,11,15,12)(10,16,18,19)(13,20,17,22)(14,21,24,23)(25,27,31,28)(26,33,32,36)(29,37,35,39)(30,38,34,40);(2,5)(6,7)(9,12,15,11)(10,16,18,19)(13,20,17,22)(14,23,24,21)(25,28,31,27)(26,33,32,36)(29,37,35,39)(30,40,34,38);(1,2)(3,6)(4,5)(7,8)(9,13,15,17)(10,14,18,24)(11,20,12,22)(16,21,19,23)(25,29,31,35)(26,34,32,30)(27,37,28,39)(33,40,36,38);(1,3)(2,6)(4,8)(5,7)(9,14)(10,17)(11,21)(12,23)(13,18)(15,24)(16,22)(19,20)(25,30,31,34)(26,35,32,29)(27,38,28,40)(33,39,36,37)

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=h,f^2=g,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, 10229433);