#7941721

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

4288 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)(3,5,7,8)(9,10)(11,20)(12,22)(13,24)(14,19)(15,16)(17,23)(18,21)(28,36)(32,39)(35,40)(37,38);(1,2)(3,5)(4,6)(7,8)(9,11,15,19)(10,14,16,20)(12,21,23,13)(17,18,22,24)(25,26,30,34)(27,29,33,31)(28,32,36,39)(35,37,40,38);(1,3,4,7)(2,5,6,8)(10,16)(11,19)(12,23)(18,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(9,12)(10,17)(11,21)(13,19)(14,18)(15,23)(16,22)(20,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(9,13,15,21)(10,18,16,24)(11,12,19,23)(14,22,20,17)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,40)(38,39);(9,14)(10,19)(11,16)(12,18)(13,17)(15,20)(21,22)(23,24)(25,29)(26,33)(27,34)(28,37)(30,31)(32,40)(35,39)(36,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,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, 7848917);