#7865071

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7772267);