#7718447

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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)(11,20)(12,13)(14,19)(15,17)(16,22)(18,23)(21,24)(28,36)(31,38)(34,40)(37,39);(1,2,5,6)(3,7,4,8)(9,11,15,22)(10,16,17,20)(12,19,18,24)(13,21,23,14)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,3,5,4)(2,7,6,8)(10,17)(12,18)(16,20)(19,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(9,12,15,18)(10,13,17,23)(11,19,22,24)(14,16,21,20);(9,13)(10,18)(11,21)(12,17)(14,22)(15,23)(16,24)(19,20)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4,5,3)(2,8,6,7)(9,14,15,21)(10,19,17,24)(11,13,22,23)(12,16,18,20)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)

Polycyclic presentation:

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

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, 7625643);