#5637444

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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

2068 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=fgh,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5544640);