#9715499

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	#59164	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4277 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9622695);