#4133797

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
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

2302 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)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26,30,31)(27,33,34,28)(29,37,38,32)(35,36,39,40);(5,6,7,8)(11,18)(13,21)(15,17)(16,20)(26,31)(27,34)(29,38)(36,40);(5,6)(7,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,27,30,34)(26,28,31,33)(29,39,38,35)(32,40,37,36);(1,2)(3,4)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,28)(26,27)(29,40)(30,33)(31,34)(32,39)(35,37)(36,38);(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,29,30,38)(26,32,31,37)(27,35,34,39)(28,36,33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4040993);