#3465214

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=g,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3372410);