#8587743

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
prime radical	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8494939);