#1302821

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2143 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=f,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,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, 1210017);