#8109903

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

4488 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)(3,4)(5,6)(7,8)(9,10)(11,17)(12,18)(13,16)(14,15)(19,23)(20,24)(21,22)(26,28)(31,34)(32,36)(38,40);(2,6)(4,7)(10,15)(11,18)(16,22)(19,24)(27,33)(31,34)(35,39)(38,40);(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(1,4,5,7)(2,3,6,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,3)(2,7)(4,6)(5,8)(9,13,14,21)(10,16,15,22)(11,19,18,24)(12,20,17,23)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|c^2=hi,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017099);