#10412407

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

4494 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

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, 10319603);