#10166232

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	#49863	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4323 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10073428);