#10444938

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

4354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=gi,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, 10352134);