#7927177

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

4305 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7834373);