#2613611

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2050 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fg,d^2=fg,e^2=fgh,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, 2520807);