#4414440

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

1766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6,8,10)(7,9,11,12)(14,18)(15,21)(20,23)(22,27)(30,35)(31,37)(33,41)(40,44);(13,14,17,18)(15,19,21,25)(16,20,24,23)(22,26,27,28)(29,30)(31,36)(32,37)(33,38)(34,35)(39,44)(40,42)(41,43);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,23)(17,18)(20,24)(21,25)(22,28)(26,27)(29,31,34,37)(30,36,35,32)(33,42,41,39)(38,40,43,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,21)(14,19,18,25)(16,22,24,27)(20,26,23,28)(29,32,34,36)(30,37,35,31)(33,40,41,44)(38,42,43,39);(5,7,8,11)(6,9,10,12)(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28)(29,33,34,41)(30,38,35,43)(31,39,37,42)(32,40,36,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=fh,d^2=f,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, 4321636);