#9850636

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

4512 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9757832);