#7643425

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

4294 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=hi,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, 7550621);