#7719945

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	#38050	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4305 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)(9,10,14,11)(12,17,19,15)(13,18,21,16)(20,22,24,23)(26,31)(29,38)(32,39)(33,36);(1,2)(3,7)(4,8)(5,6)(9,10,14,11)(12,17,19,15)(13,16,21,18)(20,23,24,22)(25,26)(27,32)(28,36)(29,34)(30,31)(33,37)(35,39)(38,40);(10,11)(12,19)(16,18)(20,24)(26,31)(28,37)(32,39)(34,40);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27,30,35)(26,32,31,39)(28,34,37,40)(29,33,38,36);(1,4,5,3)(2,8,6,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,30,37)(26,33,31,36)(27,34,35,40)(29,32,38,39);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,38)(26,34,31,40)(27,33,35,36)(28,32,37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=gh,e^2=g,f^2=i,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, 7627141);