#10368343

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

4607 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gi,d^2=hi,e^2=gh,f^2=h,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, 10275539);