#10216852

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	#38505	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4299 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=hi,e^2=ghi,f^2=ghi,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, 10124048);