#9713710

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	#46000	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4229 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=hi,f^2=gi,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, 9620906);