#10250309

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

4718 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10157505);