#9865914

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

4330 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=i,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, 9773110);