#10103451

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4263 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)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,26,30,33)(27,32,36,31)(28,29,39,37)(34,40,38,35);(1,2)(3,8)(4,6)(5,7)(10,15)(12,18)(17,21)(20,22)(25,26,30,33)(27,32,36,31)(28,37,39,29)(34,35,38,40);(1,3,5,6)(2,4,7,8)(9,11,14,16)(10,12,15,18)(13,19,23,24)(17,20,21,22)(25,27,30,36)(26,31,33,32)(28,34,39,38)(29,35,37,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,22)(23,24)(25,27)(26,32)(28,38)(29,35)(30,36)(31,33)(34,39)(37,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,16)(11,15)(13,22)(14,18)(17,19)(20,23)(21,24)(25,28,30,39)(26,29,33,37)(27,34,36,38)(31,35,32,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,23)(10,17,15,21)(11,19,16,24)(12,20,18,22)(25,29)(26,28)(27,35)(30,37)(31,34)(32,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,e^2=gh,f^2=g,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, 10010647);