#2703585

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=g,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,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, 2610781);