#7709737

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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

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, 7616933);