#3135467

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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)(9,10,13,16)(11,17,19,14)(12,15,22,21)(18,24,20,23)(25,26)(27,34)(28,30)(29,37)(31,33)(32,39)(35,40)(36,38);(6,8)(9,11,13,19)(10,14,16,17)(12,20,22,18)(15,24,21,23)(25,27,31,28)(26,30,33,34)(29,35,39,36)(32,40,37,38);(5,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,22)(21,24)(25,28,31,27)(26,30,33,34)(29,35,39,36)(32,38,37,40);(1,2,3,4)(9,10)(11,17)(12,21)(13,16)(14,19)(15,22)(18,23)(20,24)(25,29)(26,32)(27,35)(28,36)(30,40)(31,39)(33,37)(34,38);(9,12,13,22)(10,15,16,21)(11,18,19,20)(14,23,17,24)(25,30)(26,27)(28,33)(29,38)(31,34)(32,36)(35,37)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fg,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,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, 3042663);