#7657981

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

4226 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)(9,10,16,19)(11,20,15,14)(12,17,24,21)(13,22,23,18)(26,30)(31,37)(32,38)(39,40);(1,2,5,6)(3,7,4,8)(9,11,16,15)(10,14,19,20)(12,22,24,18)(13,21,23,17);(9,12)(10,17)(11,18)(13,20)(14,23)(15,22)(16,24)(19,21)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(9,13,16,23)(10,18,19,22)(11,21,15,17)(12,20,24,14);(1,3,5,4)(2,7,6,8)(9,14,16,20)(10,11,19,15)(12,23,24,13)(17,18,21,22)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(1,4,5,3)(2,8,6,7)(9,15,16,11)(10,14,19,20)(12,22,24,18)(13,17,23,21)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565177);