#9909781

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

4695 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,14)(13,16)(15,17)(18,19)(21,24)(22,23)(25,26,29,32)(27,33,35,30)(28,31,37,40)(34,39,36,38);(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)(26,30)(28,36)(29,35)(31,39)(32,33)(34,37)(38,40);(1,2)(3,5)(4,6)(7,8)(10,17)(12,19)(16,20)(21,23)(25,27,29,35)(26,30,32,33)(28,34,37,36)(31,38,40,39);(1,3)(2,5)(4,8)(6,7)(9,12,15,19)(10,18,17,14)(11,21,13,23)(16,24,20,22)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,37)(39,40);(9,13,15,11)(10,16,17,20)(12,23,19,21)(14,22,18,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(9,14,15,18)(10,19,17,12)(11,22,13,24)(16,23,20,21)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=gi,e^2=gi,f^2=gi,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, 9816977);