#10323212

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

4356 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,e^2=g,f^2=gh,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, 10230408);