#9061417

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	#57157	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4539 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8968613);