#7927249

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

4321 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)(11,19)(12,21)(13,18)(14,16)(15,20)(17,22)(23,24)(26,31)(28,34)(32,33)(35,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,19)(21,23)(22,24)(25,26,30,31)(27,33,37,32)(28,29,34,39)(35,38,40,36);(10,16)(11,20)(17,21)(18,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,33)(34,40)(38,39);(9,11)(10,15)(12,18)(13,17)(14,20)(16,19)(21,23)(22,24)(25,28,30,34)(26,29,31,39)(27,35,37,40)(32,36,33,38);(9,12,14,22)(10,17,16,21)(11,18,20,24)(13,19,23,15)(25,27)(26,33)(28,35)(29,38)(30,37)(31,32)(34,40)(36,39);(9,13)(10,18)(11,17)(12,15)(14,23)(16,24)(19,22)(20,21)(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=hi,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, 7834445);