#5280918

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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

1726 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)(7,9)(8,10)(11,12)(13,14,18,21)(15,16,23,19)(17,20,25,27)(22,24,28,26)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(6,7)(8,12)(17,25)(20,27)(22,28)(24,26)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(5,6,9,7)(8,11,12,10)(13,15,18,23)(14,19,21,16)(17,22,25,28)(20,26,27,24)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,19)(14,15,21,23)(17,24,25,26)(20,22,27,28)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,8)(6,10)(7,11)(9,12)(13,17,18,25)(14,20,21,27)(15,22,23,28)(16,24,19,26)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fgh,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=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, 5188114);