#5026103

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2238 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,18)(17,20)(22,23)(24,31)(28,30)(32,33);(13,14)(15,17)(16,18)(19,20)(21,22,26,23)(24,27,31,29)(25,28,34,30)(32,35,33,36);(5,6,8,10)(7,9,11,12)(13,14,16,18)(15,17,20,19)(21,23,26,22)(24,27,31,29)(25,28,34,30)(32,36,33,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24)(22,27)(23,29)(25,33)(26,31)(28,36)(30,35)(32,34);(5,7,8,11)(6,9,10,12)(13,15,16,20)(14,17,18,19)(21,25,26,34)(22,28,23,30)(24,32,31,33)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4933299);