#10361124

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

4465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=hi,e^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, 10268320);