#554294

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=f,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=ef

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, 461490);