#7642569

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4320 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549765);