#7724582

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	#40327	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4317 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)(11,17)(12,13)(14,16)(15,19)(18,22)(20,21)(23,24);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,20)(13,23)(14,19)(16,17)(18,21)(22,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(1,2,5,4)(3,6,8,7)(10,16)(11,19)(12,22)(21,23)(26,30)(31,37)(32,35)(38,39);(1,2,5,4)(3,6,8,7)(9,11)(10,17)(12,20)(13,21)(14,19)(15,16)(18,23)(22,24)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(9,12)(10,18)(11,20)(13,16)(14,22)(15,21)(17,23)(19,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40);(1,4,5,2)(3,7,8,6)(9,13,14,18)(10,12,16,22)(11,21,19,23)(15,24,17,20)(25,28)(26,32)(27,33)(29,36)(30,35)(31,38)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7631778);