#4457337

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2236 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)(3,4)(6,8)(10,14)(11,17)(16,21)(18,23)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(2,4)(5,6,7,8)(11,17)(15,20)(18,23)(22,24)(27,33)(30,37)(34,39)(38,40);(2,4)(5,6,7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40);(1,2)(3,4)(5,6)(7,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,c^2=i,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4364533);