#4382049

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2337 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=fg,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

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, 4289245);