#10320417

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10227613);