#7711140

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7618336);