#9712808

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3997 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)(3,4)(5,8)(6,7)(9,10,15,19)(11,18,14,16)(12,13,17,24)(20,21,22,23)(25,26,30,29)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,3)(2,4)(5,6)(7,8)(9,11,15,14)(10,16,19,18)(12,20,17,22)(13,23,24,21)(25,27,30,35)(26,31,29,33)(28,34,36,40)(32,38,37,39);(1,2,7,6)(3,5,8,4)(9,11,15,14)(10,16,19,18)(12,22,17,20)(13,21,24,23);(1,4,7,5)(2,8,6,3)(9,12,15,17)(10,13,19,24)(11,20,14,22)(16,23,18,21);(1,5,7,4)(2,8,6,3)(9,13)(10,17)(11,21)(12,19)(14,23)(15,24)(16,20)(18,22)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,6,7,2)(3,5,8,4)(9,14,15,11)(10,18,19,16)(12,22,17,20)(13,21,24,23)(25,29,30,26)(27,33,35,31)(28,32,36,37)(34,38,40,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=i,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, 9620004);