#10439967

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

4551 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=g,e^2=i,f^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, 10347163);