#9779497

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	#50683	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4239 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=g,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, 9686693);