#7782690

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

4256 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7689886);