#10058498

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

4589 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9965694);