#7743451

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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)(3,7,8,5)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,32,31)(27,37,38,33)(28,35,29,34)(30,40,39,36);(1,3,4,8)(2,5,6,7)(25,27,32,38)(26,33,31,37)(28,39,29,30)(34,36,35,40);(10,14)(15,17)(16,21)(22,23)(25,28,32,29)(26,34,31,35)(27,30,38,39)(33,40,37,36);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29,32,28)(26,35,31,34)(27,30,38,39)(33,40,37,36);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,30)(26,36)(27,28)(29,38)(31,40)(32,39)(33,35)(34,37);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,31,32,26)(27,33,38,37)(28,35,29,34)(30,40,39,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7650647);