#10196181

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

4508 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gh,e^2=g,f^2=i,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, 10103377);