#10504987

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

4395 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,21)(13,17)(15,19)(16,18)(20,24)(22,23)(25,26,30,33)(27,29,32,35)(28,31,38,40)(34,37,36,39);(1,3,5,6)(2,4,7,8)(9,11)(10,16)(12,20)(13,23)(14,19)(15,18)(17,24)(21,22)(25,27)(26,29)(28,36)(30,32)(31,39)(33,35)(34,38)(37,40);(9,12,15,13)(10,17,19,21)(11,20,18,23)(14,22,16,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,40)(35,39);(1,2)(3,8)(4,6)(5,7)(9,13,15,12)(10,17,19,21)(11,20,18,23)(14,24,16,22)(25,26)(27,35)(28,31)(29,32)(30,33)(34,39)(36,37)(38,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,14)(12,20)(13,23)(15,18)(16,19)(17,22)(21,24)(25,29,30,35)(26,32,33,27)(28,37,38,39)(31,36,40,34);(1,4)(2,6)(3,7)(5,8)(9,14)(10,18)(11,19)(12,22)(13,24)(15,16)(17,23)(20,21)(25,28)(26,31)(27,36)(29,39)(30,38)(32,34)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10412183);