#4414436

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=h,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, 4321632);