#4414449

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1814 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fh,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=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, 4321645);