#7917026

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

4338 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=h,e^2=gh,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, 7824222);