#10323654

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

4527 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=gh,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, 10230850);