#9844102

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

4482 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)(11,14)(12,15)(13,18)(16,21)(17,20)(19,23)(22,24)(25,26,32,29)(27,30,33,28)(31,37,39,34)(35,36,40,38);(1,2,5,7)(3,4,8,6)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,27)(26,30)(28,29)(31,35)(32,33)(34,38)(36,37)(39,40);(1,3)(2,6)(4,7)(5,8)(10,11)(15,17)(16,18)(22,23)(25,28,32,30)(26,33,29,27)(31,36,39,38)(34,40,37,35);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,29,32,26)(27,28,33,30)(31,37,39,34)(35,36,40,38);(1,4,5,6)(2,3,7,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,30,32,28)(26,33,29,27)(31,36,39,38)(34,35,37,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,31,32,39)(26,34,29,37)(27,35,33,40)(28,36,30,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=h,e^2=gi,f^2=gh,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, 9751298);