#10368628

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

4487 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)(3,4)(5,6)(7,8)(9,10,16,19)(11,20,14,17)(12,15,13,18)(21,22,23,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(2,6)(3,8)(9,11,16,14)(10,17,19,20)(12,23,13,21)(15,22,18,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(2,6)(4,7)(9,12,16,13)(10,18,19,15)(11,21,14,23)(17,22,20,24)(25,26,29,32)(27,30,35,33)(28,31,37,36)(34,38,40,39);(1,3)(2,7)(4,6)(5,8)(9,13,16,12)(10,15,19,18)(11,21,14,23)(17,22,20,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4,5,7)(2,3,6,8)(9,14,16,11)(10,17,19,20)(12,23,13,21)(15,24,18,22)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3)(2,7)(4,6)(5,8)(9,15)(10,12)(11,22)(13,19)(14,24)(16,18)(17,23)(20,21)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=ghi,e^2=g,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, 10275824);