#5609282

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1724 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)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,27)(25,26)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(5,7,8,11)(6,9,10,12)(14,18)(19,21)(20,26)(27,28)(31,38)(33,41)(35,37)(36,40);(5,7)(6,9)(8,11)(10,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,27)(25,26)(29,31,34,38)(30,35,32,37)(33,39,41,44)(36,42,40,43);(1,2)(3,4)(13,15,17,23)(14,19,18,21)(16,24,25,22)(20,28,26,27)(29,32,34,30)(31,35,38,37)(33,40,41,36)(39,42,44,43);(5,6)(7,9)(8,10)(11,12)(13,16,17,25)(14,20,18,26)(15,22,23,24)(19,27,21,28)(29,33)(30,36)(31,39)(32,40)(34,41)(35,42)(37,43)(38,44)

Polycyclic presentation:

2^9|a^2=fh,b^2=fgh,c^2=g,d^2=fg,e^2=f,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, 5516478);