#10103458

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

4281 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)(11,20)(12,18)(13,14)(15,17)(16,19)(21,23)(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)(9,11,16,15)(10,17,19,20)(12,21,13,22)(14,23,18,24)(25,26,30,33)(27,32,36,31)(28,37,39,29)(34,35,38,40);(1,3,5,6)(2,4,7,8)(9,12,16,13)(10,18,19,14)(11,21,15,22)(17,24,20,23)(25,27,30,36)(26,31,33,32)(28,34,39,38)(29,35,37,40);(1,4)(2,6)(3,7)(5,8)(9,13,16,12)(10,18,19,14)(11,22,15,21)(17,24,20,23)(25,27)(26,32)(28,38)(29,35)(30,36)(31,33)(34,39)(37,40);(1,2)(3,4)(5,7)(6,8)(9,14,16,18)(10,13,19,12)(11,23,15,24)(17,22,20,21)(25,28,30,39)(26,29,33,37)(27,34,36,38)(31,35,32,40);(1,4)(2,6)(3,7)(5,8)(9,15,16,11)(10,17,19,20)(12,22,13,21)(14,23,18,24)(25,29)(26,28)(27,35)(30,37)(31,34)(32,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,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, 10010654);