#10106199

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

4721 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,21)(22,23)(25,26,30,33)(27,31,37,40)(28,29,38,32)(34,39,35,36);(1,2,4,6)(3,5,8,7)(9,11)(10,13)(12,20)(14,18)(15,17)(16,24)(19,22)(21,23)(25,27,30,37)(26,31,33,40)(28,35,38,34)(29,36,32,39);(1,2)(3,5)(4,6)(7,8)(10,15)(11,18)(16,23)(19,20)(25,28,30,38)(26,32,33,29)(27,34,37,35)(31,36,40,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,22)(23,24)(25,28)(26,29)(27,35)(30,38)(31,36)(32,33)(34,37)(39,40);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,27)(26,31)(28,34)(29,39)(30,37)(32,36)(33,40)(35,38);(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,23)(22,24)(25,29)(26,28)(27,36)(30,32)(31,35)(33,38)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,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, 10013395);