#10322384

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
prime radical	#47508	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4263 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)(11,16)(12,14)(13,17)(15,19)(18,23)(20,24)(21,22)(25,26,32,36)(27,37,39,33)(28,31,34,29)(30,35,40,38);(1,2)(3,7)(4,5)(6,8)(9,11,15,23)(10,16,19,18)(12,21,13,20)(14,22,17,24)(25,27,32,39)(26,33,36,37)(28,30,34,40)(29,35,31,38);(2,5)(6,7)(9,12,15,13)(10,17,19,14)(11,20,23,21)(16,22,18,24)(25,28)(26,29)(27,30)(31,36)(32,34)(33,35)(37,38)(39,40);(2,5)(6,7)(9,13,15,12)(10,17,19,14)(11,21,23,20)(16,22,18,24)(25,29,32,31)(26,34,36,28)(27,38,39,35)(30,37,40,33);(1,2)(3,6)(4,5)(7,8)(9,14,15,17)(10,12,19,13)(11,22,23,24)(16,21,18,20)(25,30)(26,35)(27,34)(28,39)(29,37)(31,33)(32,40)(36,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,18)(12,20)(13,21)(14,22)(15,23)(16,19)(17,24)(25,31,32,29)(26,28,36,34)(27,35,39,38)(30,33,40,37)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=h,e^2=gh,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, 10229580);