#10103404

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	#38521	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4213 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)(11,16)(12,14)(13,17)(15,19)(18,24)(20,22)(21,23)(25,26,31,33)(27,36,38,32)(28,39,35,30)(29,34,40,37);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,16)(13,18)(14,19)(17,23)(20,24)(21,22)(25,27,31,38)(26,32,33,36)(28,37,35,34)(29,39,40,30);(1,3,5,6)(2,4,7,8)(26,33)(28,35)(32,36)(34,37);(1,4)(2,6)(3,7)(5,8)(9,12,14,10)(11,16,19,15)(13,20,22,17)(18,24,21,23)(25,28,31,35)(26,30,33,39)(27,37,38,34)(29,36,40,32);(1,2)(3,4)(5,7)(6,8)(9,11)(10,16)(12,15)(13,21)(14,19)(17,23)(18,22)(20,24)(25,29,31,40)(26,34,33,37)(27,30,38,39)(28,32,35,36);(1,4)(2,6)(3,7)(5,8)(9,13)(10,17)(11,18)(12,20)(14,22)(15,23)(16,24)(19,21)(25,30)(26,35)(27,29)(28,33)(31,39)(32,34)(36,37)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=g,e^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, 10010600);