#9788986

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

4318 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=g,e^2=g,f^2=h,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, 9696182);