#10412007

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

4541 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)(3,7,8,5)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,24)(20,23)(26,30)(27,33)(28,35)(39,40);(1,2,4,6)(3,5,8,7)(10,15)(12,22)(16,18)(19,23)(25,26,29,30)(27,31,33,37)(28,32,35,38)(34,39,36,40);(1,2)(3,5)(4,6)(7,8)(25,27)(26,31)(28,34)(29,33)(30,37)(32,39)(35,36)(38,40);(3,8)(5,7)(9,11)(10,16)(12,23)(13,24)(14,21)(15,18)(17,20)(19,22)(25,27)(26,31)(28,36)(29,33)(30,37)(32,40)(34,35)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,19)(13,15)(14,22)(16,24)(18,20)(21,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,38)(36,40);(1,3)(2,5)(4,8)(6,7)(9,13,14,17)(10,12,15,22)(11,20,21,24)(16,19,18,23)(25,28,29,35)(26,32,30,38)(27,34,33,36)(31,39,37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,f^2=g,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, 10319203);