#3464168

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2193 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)(10,11)(12,19)(13,21)(23,24)(25,26,29,33)(27,31,35,30)(28,32,38,37)(34,40,36,39);(6,8)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,23,22,24)(25,27,29,35)(26,30,33,31)(28,34,38,36)(32,39,37,40);(5,6)(7,8)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,23,22,24)(25,27)(26,31)(28,36)(29,35)(30,33)(32,39)(34,38)(37,40);(1,2,3,4)(9,12)(10,15)(11,17)(13,22)(14,19)(16,24)(18,23)(20,21)(25,26)(27,31)(28,37)(29,33)(30,35)(32,38)(34,39)(36,40);(9,13,14,21)(10,16,11,18)(12,20,19,22)(15,23,17,24)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,37)(35,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=g,d^2=fi,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, 3371364);