#1403654

order	512
nilpotency class	2
isoclinism	isoclinic class of #1393169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fhi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1310850);