#9779470

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
prime radical	#45563	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4489 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=h,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, 9686666);